2.03b Probability diagrams: tree, Venn, sample space

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 \(A\).

4 Two fair dice are thrown.

1. Event \(A\) is 'the scores differ by 3 or more'. Find the probability of event \(A\).
2. Event \(B\) is 'the product of the scores is greater than 8 '. Find the probability of event \(B\).
3. State with a reason whether events \(A\) and \(B\) are mutually exclusive.

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 \(X\).

6 There are three sets of traffic lights on Karinne's journey to work. The independent probabilities that Karinne has to stop at the first, second and third set of lights are \(0.4,0.8\) and 0.3 respectively.

1. Draw a tree diagram to show this information.
2. Find the probability that Karinne has to stop at each of the first two sets of lights but does not have to stop at the third set.
3. Find the probability that Karinne has to stop at exactly two of the three sets of lights.
4. Find the probability that Karinne has to stop at the first set of lights, given that she has to stop at exactly two sets of lights.

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.

5 Three friends, Rick, Brenda and Ali, go to a football match but forget to say which entrance to the ground they will meet at. There are four entrances, \(A , B , C\) and \(D\). Each friend chooses an entrance independently.

• The probability that Rick chooses entrance \(A\) is \(\frac { 1 } { 3 }\). The probabilities that he chooses entrances \(B , C\) or \(D\) are all equal.
• Brenda is equally likely to choose any of the four entrances.
• The probability that Ali chooses entrance \(C\) is \(\frac { 2 } { 7 }\) and the probability that he chooses entrance \(D\) is \(\frac { 3 } { 5 }\). The probabilities that he chooses the other two entrances are equal.
  1. Find the probability that at least 2 friends will choose entrance \(B\).
  2. Find the probability that the three friends will all choose the same entrance.

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.

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.

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]

7 James has a fair coin and a fair tetrahedral die with four faces numbered 1, 2, 3, 4. He tosses the coin once and the die twice. The random variable \(X\) is defined as follows.

• If the coin shows a head then \(X\) is the sum of the scores on the two throws of the die.
• If the coin shows a tail then \(X\) is the score on the first throw of the die only.
  1. Explain why \(X = 1\) can only be obtained by throwing a tail, and show that \(\mathrm { P } ( X = 1 ) = \frac { 1 } { 8 }\).
  2. Show that \(\mathrm { P } ( X = 3 ) = \frac { 3 } { 16 }\).
  3. Copy and complete the probability distribution table for \(X\).

Event \(Q\) is 'James throws a tail'. Event \(R\) is 'the value of \(X\) is 7'.

Determine whether events \(Q\) and \(R\) are exclusive. Justify your answer.

6 Nadia is very forgetful. Every time she logs in to her online bank she only has a \(40 \%\) chance of remembering her password correctly. She is allowed 3 unsuccessful attempts on any one day and then the bank will not let her try again until the next day.

1. Draw a fully labelled tree diagram to illustrate this situation.
2. Let \(X\) be the number of unsuccessful attempts Nadia makes on any day that she tries to log in to her bank. Copy and complete the following table to show the probability distribution of \(X\).

   \(x\)	0	1	2	3
   \(\mathrm { P } ( X = x )\)		0.24

3. Calculate the expected number of unsuccessful attempts made by Nadia on any day that she tries to \(\log\) in.

3 Ellie throws two fair tetrahedral dice, each with faces numbered 1, 2, 3 and 4. She notes the numbers on the faces that the dice land on. Event \(S\) is 'the sum of the two numbers is 4 '. Event \(T\) is 'the product of the two numbers is an odd number'.

1. Determine whether events \(S\) and \(T\) are independent, showing your working.
2. Are events \(S\) and \(T\) exclusive? Justify your answer.

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.

4 For a group of 250 cars the numbers, classified by colour and country of manufacture, are shown in the table. One car is selected at random from this group. Find the probability that the selected car is

1. a red or silver car manufactured in Korea,
2. not manufactured in Japan. \(X\) is the event that the selected car is white. \(Y\) is the event that the selected car is manufactured in Germany.
3. By using appropriate probabilities, determine whether events \(X\) and \(Y\) are independent.

3 A box contains 3 red balls and 5 blue balls. One ball is taken at random from the box and not replaced. A yellow ball is then put into the box. A second ball is now taken at random from the box.

1. Complete the tree diagram to show all the outcomes and the probability for each branch. First ball
   Second ball \includegraphics[max width=\textwidth, alt={}, center]{7dc85f33-2647-4f73-8093-524b70f99767-04_655_392_688_474} \includegraphics[max width=\textwidth, alt={}, center]{7dc85f33-2647-4f73-8093-524b70f99767-04_785_387_703_1110}
2. Find the probability that the two balls taken are the same colour.
3. Find the probability that the first ball taken is red, given that the second ball taken is blue.

1 When Shona goes to college she either catches the bus with probability 0.8 or she cycles with probability 0.2 . If she catches the bus, the probability that she is late is 0.4 . If she cycles, the probability that she is late is \(x\). The probability that Shona is not late for college on a randomly chosen day is 0.63 . Find the value of \(x\).

1 There are 300 students at a music college. All students play exactly one of the guitar, the piano or the flute. The numbers of male and female students that play each of the instruments are given in the following table.

1. Find the probability that a randomly chosen student at the college is a male who does not play the piano.
2. Determine whether the events 'a randomly chosen student is male' and 'a randomly chosen student does not play the piano' are independent, justifying your answer.

6 Nadia is very forgetful. Every time she logs in to her online bank she only has a \(40 \%\) chance of remembering her password correctly. She is allowed 3 unsuccessful attempts on any one day and then the bank will not let her try again until the next day.

1. Draw a fully labelled tree diagram to illustrate this situation.
2. Let \(X\) be the number of unsuccessful attempts Nadia makes on any day that she tries to log in to her bank. Complete the following table to show the probability distribution of \(X\).

   \(x\)	0	1	2	3
   \(\mathrm { P } ( X = x )\)		0.24

3. Calculate the expected number of unsuccessful attempts made by Nadia on any day that she tries to \(\log\) in.

3 It was found that \(68 \%\) of the passengers on a train used a cell phone during their train journey. Of those using a cell phone, \(70 \%\) were under 30 years old, \(25 \%\) were between 30 and 65 years old and the rest were over 65 years old. Of those not using a cell phone, \(26 \%\) were under 30 years old and \(64 \%\) were over 65 years old.

1. Draw a tree diagram to represent this information, giving all probabilities as decimals.
2. Given that one of the passengers is 45 years old, find the probability of this passenger using a cell phone during the journey.

1 Marie wants to choose one student at random from Anthea, Bill and Charlie. She throws two fair coins. If both coins show tails she will choose Anthea. If both coins show heads she will choose Bill. If the coins show one of each she will choose Charlie.

1. Explain why this is not a fair method for choosing the student.
2. Describe how Marie could use the two coins to give a fair method for choosing the student.

7 A bag contains three 1 p coins and seven 2 p coins. Coins are removed at random one at a time, without replacement, until the total value of the coins removed is at least 3p. Then no more coins are removed.

1. Copy and complete the probability tree diagram. First coin \includegraphics[max width=\textwidth, alt={}, center]{43f7e091-9ae7-4373-a209-e2ebdba5260f-4_350_317_1279_568} Find the probability that
2. exactly two coins are removed,
3. the total value of the coins removed is 4p.

6 Two bags contain coloured discs. At first, bag \(P\) contains 2 red discs and 2 green discs, and bag \(Q\) contains 3 red discs and 1 green disc. A disc is chosen at random from bag \(P\), its colour is noted and it is placed in bag \(Q\). A disc is then chosen at random from bag \(Q\), its colour is noted and it is placed in bag \(P\). A disc is then chosen at random from bag \(P\). The tree diagram shows the different combinations of three coloured discs chosen. \includegraphics[max width=\textwidth, alt={}, center]{5faf0d93-4037-4958-8665-1008477a79de-5_863_986_559_612}

1. Write down the values of \(a , b , c , d , e\) and \(f\). The total number of red discs chosen, out of 3, is denoted by \(R\). The table shows the probability distribution of \(R\).

   \(r\)	0	1	2	3
   \(\mathrm { P } ( R = r )\)	\(\frac { 1 } { 10 }\)	\(k\)	\(\frac { 9 } { 20 }\)	\(\frac { 1 } { 5 }\)

2. Show how to obtain the value \(\mathrm { P } ( R = 2 ) = \frac { 9 } { 20 }\).
3. Find the value of \(k\).
4. Calculate the mean and variance of \(R\).

5 A school athletics team has 10 members. The table shows which competitions each of the members can take part in. An athlete is selected at random. Events \(A , B , C , D\) are defined as follows.
A: the athlete can take part in exactly 2 competitions. \(B\) : the athlete can take part in the 200 m . \(C\) : the athlete can take part in the 110 m hurdles. \(D\) : the athlete can take part in the long jump.

1. Write down the value of \(\mathrm { P } ( A \cap B )\).
2. Write down the value of \(\mathrm { P } ( C \cup D )\).
3. Which two of the four events \(A , B , C , D\) are mutually exclusive?
4. Show that events \(B\) and \(D\) are not independent.