2.03d Calculate conditional probability: from first principles

1 On each day that Tamar goes to work, he wears either a blue suit with probability 0.6 or a grey suit with probability 0.4 . If he wears a blue suit then the probability that he wears red socks is 0.2 . If he wears a grey suit then the probability that he wears red socks is 0.32 .

1. Find the probability that Tamar wears red socks on any particular day that he is at work.
2. Given that Tamar is not wearing red socks at work, find the probability that he is wearing a grey suit.

5 Rachel and Anna play each other at badminton. Each game results in either a win for Rachel or a win for Anna. The probability of Rachel winning the first game is 0.6 . If Rachel wins a particular game, the probability of her winning the next game is 0.7 , but if she loses, the probability of her winning the next game is 0.4 . By using a tree diagram, or otherwise,

1. find the conditional probability that Rachel wins the first game, given that she loses the second,
2. find the probability that Rachel wins 2 games and loses 1 game out of the first three games they play.

5 In a certain country \(54 \%\) of the population is male. It is known that \(5 \%\) of the males are colour-blind and \(2 \%\) of the females are colour-blind. A person is chosen at random and found to be colour-blind. By drawing a tree diagram, or otherwise, find the probability that this person is male.

3 When Andrea needs a taxi, she rings one of three taxi companies, A, B or C. 50\% of her calls are to taxi company \(A , 30 \%\) to \(B\) and \(20 \%\) to \(C\). A taxi from company \(A\) arrives late \(4 \%\) of the time, a taxi from company \(B\) arrives late \(6 \%\) of the time and a taxi from company \(C\) arrives late \(17 \%\) of the time.

1. Find the probability that, when Andrea rings for a taxi, it arrives late.
2. Given that Andrea's taxi arrives late, find the conditional probability that she rang company \(B\).

2 Boxes of sweets contain toffees and chocolates. Box \(A\) contains 6 toffees and 4 chocolates, box \(B\) contains 5 toffees and 3 chocolates, and box \(C\) contains 3 toffees and 7 chocolates. One of the boxes is chosen at random and two sweets are taken out, one after the other, and eaten.

1. Find the probability that they are both toffees.
2. Given that they are both toffees, find the probability that they both came from box \(A\).

7 Box \(A\) contains 5 red paper clips and 1 white paper clip. Box \(B\) contains 7 red paper clips and 2 white paper clips. One paper clip is taken at random from box \(A\) and transferred to box \(B\). One paper clip is then taken at random from box \(B\).

1. Find the probability of taking both a white paper clip from box \(A\) and a red paper clip from box \(B\).
2. Find the probability that the paper clip taken from box \(B\) is red.
3. Find the probability that the paper clip taken from box \(A\) was red, given that the paper clip taken from box \(B\) is red.
4. The random variable \(X\) denotes the number of times that a red paper clip is taken. Draw up a table to show the probability distribution of \(X\).

3 Maria chooses toast for her breakfast with probability 0.85 . If she does not choose toast then she has a bread roll. If she chooses toast then the probability that she will have jam on it is 0.8 . If she has a bread roll then the probability that she will have jam on it is 0.4 .

1. Draw a fully labelled tree diagram to show this information.
2. Given that Maria did not have jam for breakfast, find the probability that she had toast.

7 Sanket plays a game using a biased die which is twice as likely to land on an even number as on an odd number. The probabilities for the three even numbers are all equal and the probabilities for the three odd numbers are all equal.

1. Find the probability of throwing an odd number with this die. Sanket throws the die once and calculates his score by the following method.
   • If the number thrown is 3 or less he multiplies the number thrown by 3 and adds 1 .
   • If the number thrown is more than 3 he multiplies the number thrown by 2 and subtracts 4 .
   The random variable \(X\) is Sanket's score.
2. Show that \(\mathrm { P } ( X = 8 ) = \frac { 2 } { 9 }\). The table shows the probability distribution of \(X\).

   \(x\)	4	6	7	8	10
   \(\mathrm { P } ( X = x )\)	\(\frac { 3 } { 9 }\)	\(\frac { 1 } { 9 }\)	\(\frac { 2 } { 9 }\)	\(\frac { 2 } { 9 }\)	\(\frac { 1 } { 9 }\)

3. Given that \(\mathrm { E } ( X ) = \frac { 58 } { 9 }\), find \(\operatorname { Var } ( X )\). Sanket throws the die twice.
4. Find the probability that the total of the scores on the two throws is 16 .
5. Given that the total of the scores on the two throws is 16 , find the probability that the score on the first throw was 6 .

3 A fair five-sided spinner has sides numbered 1,2,3,4,5. Raj spins the spinner and throws two fair dice. He calculates his score as follows.

• If the spinner lands on an even-numbered side, Raj multiplies the two numbers showing on the dice to get his score.
• If the spinner lands on an odd-numbered side, Raj adds the numbers showing on the dice to get his score.

Given that Raj's score is 12, find the probability that the spinner landed on an even-numbered side.

7 A committee of 6 people, which must contain at least 4 men and at least 1 woman, is to be chosen from 10 men and 9 women.

1. Find the number of possible committees that can be chosen.
2. Find the probability that one particular man, Albert, and one particular woman, Tracey, are both on the committee.
3. Find the number of possible committees that include either Albert or Tracey but not both.
4. The committee that is chosen consists of 4 men and 2 women. They queue up randomly in a line for refreshments. Find the probability that the women are not next to each other in the queue.

1 Fabio drinks coffee each morning. He chooses Americano, Cappucino or Latte with probabilities 0.5, 0.3 and 0.2 respectively. If he chooses Americano he either drinks it immediately with probability 0.8 , or leaves it to drink later. If he chooses Cappucino he either drinks it immediately with probability 0.6 , or leaves it to drink later. If he chooses Latte he either drinks it immediately with probability 0.1 , or leaves it to drink later.

1. Find the probability that Fabio chooses Americano and leaves it to drink later.
2. Fabio drinks his coffee immediately. Find the probability that he chose Latte.

3 Ronnie obtained data about the gross domestic product (GDP) and the birth rate for 170 countries. He classified each GDP and each birth rate as either 'low', 'medium' or 'high'. The table shows the number of countries in each category. One of these countries is chosen at random.

1. Find the probability that the country chosen has a medium GDP.
2. Find the probability that the country chosen has a low birth rate, given that it does not have a medium GDP.
3. State with a reason whether or not the events 'the country chosen has a high GDP' and 'the country chosen has a high birth rate' are exclusive. One country is chosen at random from those countries which have a medium GDP and then a different country is chosen at random from those which have a medium birth rate.
4. Find the probability that both countries chosen have a medium GDP and a medium birth rate.

2 The people living in two towns, Mumbok and Bagville, are classified by age. The numbers in thousands living in each town are shown in the table below. One of the towns is chosen. The probability of choosing Mumbok is 0.6 and the probability of choosing Bagville is 0.4 . Then a person is chosen at random from that town. Given that the person chosen is between 18 and 60 years old, find the probability that the town chosen was Mumbok. [5]

2 On Saturday afternoons Mohit goes shopping with probability 0.25, or goes to the cinema with probability 0.35 or stays at home. If he goes shopping the probability that he spends more than \(\\) 50\( is 0.7 . If he goes to the cinema the probability that he spends more than \)\\( 50\) is 0.8 . If he stays at home he spends \(\\) 10$ on a pizza.

1. Find the probability that Mohit will go to the cinema and spend less than \(\\) 50\(.
2. Given that he spends less than \)\\( 50\), find the probability that he went to the cinema.

3 Jodie tosses a biased coin and throws two fair tetrahedral dice. The probability that the coin shows a head is \(\frac { 1 } { 3 }\). Each of the dice has four faces, numbered \(1,2,3\) and 4 . Jodie's score is calculated from the numbers on the faces that the dice land on, as follows:

• if the coin shows a head, the two numbers from the dice are added together;
• if the coin shows a tail, the two numbers from the dice are multiplied together.

Find the probability that the coin shows a head given that Jodie's score is 8 .

7 A box contains 2 green apples and 2 red apples. Apples are taken from the box, one at a time, without replacement. When both red apples have been taken, the process stops. The random variable \(X\) is the number of apples which have been taken when the process stops.

1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 1 } { 3 }\).
2. Draw up the probability distribution table for \(X\). Another box contains 2 yellow peppers and 5 orange peppers. Three peppers are taken at random from the box without replacement.
3. Given that at least 2 of the peppers taken from the box are orange, find the probability that all 3 peppers are orange.

6 A fair spinner \(A\) has edges numbered \(1,2,3,3\). A fair spinner \(B\) has edges numbered \(- 3 , - 2 , - 1,1\). Each spinner is spun. The number on the edge that the spinner comes to rest on is noted. Let \(X\) be the sum of the numbers for the two spinners.

1. Copy and complete the table showing the possible values of \(X\).

   	Spinner \(A\)
   \cline { 2 - 6 }		1	2	3	3
   Spinner \(B\)	- 2
   - 2			1
   - 1
   1

2. Draw up a table showing the probability distribution of \(X\).
3. Find \(\operatorname { Var } ( X )\).
4. Find the probability that \(X\) is even, given that \(X\) is positive.

2 In country \(X , 25 \%\) of people have fair hair. In country \(Y , 60 \%\) of people have fair hair. There are 20 million people in country \(X\) and 8 million people in country \(Y\). A person is chosen at random from these 28 million people.

1. Find the probability that the person chosen is from country \(X\).
2. Find the probability that the person chosen has fair hair.
3. Find the probability that the person chosen is from country \(X\), given that the person has fair hair.

6 Deeti has 3 red pens and 1 blue pen in her left pocket and 3 red pens and 1 blue pen in her right pocket. 'Operation \(T\) ' consists of Deeti taking one pen at random from her left pocket and placing it in her right pocket, then taking one pen at random from her right pocket and placing it in her left pocket.

1. Find the probability that, when Deeti carries out operation \(T\), she takes a blue pen from her left pocket and then a blue pen from her right pocket. The random variable \(X\) is the number of blue pens in Deeti's left pocket after carrying out operation \(T\).
2. Find \(\mathrm { P } ( X = 1 )\).
3. Given that the pen taken from Deeti's right pocket is blue, find the probability that the pen taken from Deeti's left pocket is blue.

1 When Anya goes to school, the probability that she walks is 0.3 and the probability that she cycles is 0.65 ; if she does not walk or cycle she takes the bus. When Anya walks the probability that she is late is 0.15 . When she cycles the probability that she is late is 0.1 and when she takes the bus the probability that she is late is 0.6 . Given that Anya is late, find the probability that she cycles.

5 Over a period of time Julian finds that on long-distance flights he flies economy class on \(82 \%\) of flights. On the rest of the flights he flies first class. When he flies economy class, the probability that he gets a good night's sleep is \(x\). When he flies first class, the probability that he gets a good night's sleep is 0.9 .

1. Draw a fully labelled tree diagram to illustrate this situation. The probability that Julian gets a good night's sleep on a randomly chosen flight is 0.285 .
2. Find the value of \(x\).
3. Given that on a particular flight Julian does not get a good night's sleep, find the probability that he is flying economy class.

3 At the end of a revision course in mathematics, students have to pass a test to gain a certificate. The probability of any student passing the test at the first attempt is 0.85 . Those students who fail are allowed to retake the test once, and the probability of any student passing the retake test is 0.65 .

1. Draw a fully labelled tree diagram to show all the outcomes.
2. Given that a student gains the certificate, find the probability that this student fails the test on the first attempt.

2 Benju cycles to work each morning and he has two possible routes. He chooses the hilly route with probability 0.4 and the busy route with probability 0.6 . If he chooses the hilly route, the probability that he will be late for work is \(x\) and if he chooses the busy route the probability that he will be late for work is \(2 x\). The probability that Benju is late for work on any day is 0.36 .

1. Show that \(x = 0.225\).
2. Given that Benju is not late for work, find the probability that he chooses the hilly route.

4 The random variable \(A\) has the distribution \(\operatorname { Po } ( 1.5 ) . A _ { 1 }\) and \(A _ { 2 }\) are independent values of \(A\).

1. Find \(\mathrm { P } \left( A _ { 1 } + A _ { 2 } < 2 \right)\).
2. Given that \(A _ { 1 } + A _ { 2 } < 2\), find \(\mathrm { P } \left( A _ { 1 } = 1 \right)\).
3. Give a reason why \(A _ { 1 } - A _ { 2 }\) cannot have a Poisson distribution.

5 The number of goals scored by a sports team in the first half of any match has the distribution \(X \sim \mathrm { Po }\) (3.1). The number of goals scored by the same team in the second half of any match has the distribution \(Y \sim \operatorname { Po } ( 2.4 )\). You may assume that the distributions of \(X\) and \(Y\) are independent.

1. Find \(\mathrm { P } ( X < 4 )\).
2. Find the probability that, in a randomly chosen match, the team scores at least 5 goals.
3. Given that the team scores a total of 5 goals in a randomly chosen match, find the probability that they score exactly 3 goals in the first half.