2.03d Calculate conditional probability: from first principles

4

1. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\). Prove that the probability generating function, \(\mathrm { G } _ { X } ( t )\), is given by $$\mathrm { G } _ { X } ( t ) = \mathrm { e } ^ { \lambda ( t - 1 ) } .$$
2. The independent random variables \(X\) and \(Y\) have distributions \(\operatorname { Po } ( \lambda )\) and \(\operatorname { Po } ( \mu )\) respectively. Use probability generating functions to show that the distribution of \(X + Y\) is \(\operatorname { Po } ( \lambda + \mu )\).
3. Given that \(X \sim \operatorname { Po } ( 1.5 )\) and \(Y \sim \operatorname { Po } ( 2.5 )\), find \(\mathrm { P } ( X \leqslant 2 \mid X + Y = 4 )\).

4 At a sixth form college, the student council has 16 members made up as follows. There are 3 male and 3 female students from Year 12, and 6 male and 4 female students from Year 13. Two members of the council are chosen at random to represent the college at conference. Find the probability that the 2 members chosen are

1. the same sex,
2. the same sex and from the same year,
3. from the same year given that they are the same sex.

3 The probability distribution of the discrete random variable \(X\) is defined as follows. $$\mathrm { P } ( X = x ) = k ( 2 + x ) ( 5 - x ) \quad \text { for } x = 0,1,2,3,4$$

1. Show that \(k = \frac { 1 } { 50 }\).
2. Find the variance of \(X\).
3. Find \(\mathrm { P } ( X = 4 \mid X > 0 )\).

10

1. \(X , Y\) and \(Z\) are independent random variables having Poisson distributions with means \(\lambda , \mu\) and \(\lambda + \mu\) respectively. Find \(\mathrm { P } ( X = 0\) and \(Y = 2 ) , \mathrm { P } ( X = 1\) and \(Y = 1 )\) and \(\mathrm { P } ( X = 2\) and \(Y = 0 )\). Hence verify that \(\mathrm { P } ( X + Y = 2 ) = \mathrm { P } ( Z = 2 )\).
2. In an office the male absence rate, i.e. the number of working days lost each month due to the absence of male employees, has a Poisson distribution with mean 4.5. In the same office the female absence rate has an independent Poisson distribution with mean 4.1. Calculate the probability that
   1. during a particular month both the male absence rate and the female absence rate are equal to 3,
   2. during a particular month the total of the male and female absence rates is equal to 6,
   3. during a particular month the male and female absence rates were each equal to 3 , given that the total of the male and female absence rates was equal to 6 .

A small farm has 5 ducks and 2 geese. Four of these birds are to be chosen at random. The random variable \(X\) represents the number of geese chosen.

1. Draw up the probability distribution of \(X\). [3]
2. Show that \(\mathrm{E}(X) = \frac{8}{7}\) and calculate \(\mathrm{Var}(X)\). [3]
3. When the farmer's dog is let loose, it chases either the ducks with probability \(\frac{3}{5}\) or the geese with probability \(\frac{2}{5}\). If the dog chases the ducks there is a probability of \(\frac{1}{10}\) that they will attack the dog. If the dog chases the geese there is a probability of \(\frac{1}{4}\) that they will attack the dog. Given that the dog is not attacked, find the probability that it was chasing the geese. [4]

Nikita goes shopping to buy a birthday present for her mother. She buys either a scarf, with probability 0.3, or a handbag. The probability that her mother will like the choice of scarf is 0.72. The probability that her mother will like the choice of handbag is \(x\). This information is shown on the tree diagram. The probability that Nikita's mother likes the present that Nikita buys is 0.783.

1. Find \(x\). [3]
2. Given that Nikita's mother does not like her present, find the probability that the present is a scarf. [4]

A bag contains a large number of coloured counters. Each counter is labelled A, B or C 30% of the counters are labelled A 45% of the counters are labelled B The rest of the counters are labelled C It is known that 2% of the counters labelled A are red 4% of the counters labelled B are red 6% of the counters labelled C are red One counter is selected at random from the bag.

1. Complete the tree diagram on the opposite page to illustrate this information. [2]
2. Calculate the probability that the counter is labelled A and is not red. [2]
3. Calculate the probability that the counter is red. [2]
4. Given that the counter is red, find the probability that it is labelled C [3]

\includegraphics{figure_3}

Jake and Kamil are sometimes late for school. The events \(J\) and \(K\) are defined as follows \(J =\) the event that Jake is late for school \(K =\) the event that Kamil is late for school \(\text{P}(J) = 0.25\), \(\text{P}(J \cap K) = 0.15\) and \(\text{P}(J' \cap K') = 0.7\) On a randomly selected day, find the probability that

1. at least one of Jake or Kamil are late for school, [1]
2. Kamil is late for school. [2]

Given that Jake is late for school,

1. find the probability that Kamil is late. [3]

The teacher suspects that Jake being late for school and Kamil being late for school are linked in some way.

1. Determine whether or not \(J\) and \(K\) are statistically independent. [2]
2. Comment on the teacher's suspicion in the light of your calculation in (d). [1]

The events \(A\) and \(B\) are independent such that \(P(A) = 0.25\) and \(P(B) = 0.30\). Find

1. \(P(A \cap B)\), [2]
2. \(P(A \cup B)\), [2]
3. \(P(A | B')\). [4]

For any married couple who are members of a tennis club, the probability that the husband has a degree is \(\frac{3}{5}\) and the probability that the wife has a degree is \(\frac{1}{2}\). The probability that the husband has a degree, given that the wife has a degree, is \(\frac{11}{12}\). A married couple is chosen at random.

1. Show that the probability that both of them have degrees is \(\frac{11}{24}\). [2]
2. Draw a Venn diagram to represent these data. [5]

Find the probability that

1. only one of them has a degree, [2]
2. neither of them has a degree. [3]

Two married couples are chosen at random.

1. Find the probability that only one of the two husbands and only one of the two wives have degrees. [6]

The values of the two variables \(A\) and \(B\) given in the table in Question 5 are written on cards and placed in two separate packs, which are labelled \(A\) and \(B\). One card is selected from Pack \(A\). Let \(A_i\) represent the event that the first digit on this card is \(i\).

1. Write down the value of P\((A_2)\). [1 mark] The card taken from Pack \(A\) is now transferred into Pack \(B\), and one card is picked at random from Pack \(B\). Let \(B_i\) represent the event that the first digit on this card is \(i\).
2. Show that P\((A_1 \cap B_1) = \frac{1}{24}\). [3 marks]
3. Show that P\((A_6 | B_2) = \frac{4}{41}\). [5 marks]
4. Find the value of P\((A_1 \cup B_4)\). [5 marks]

Sixteen cards have been lost from a pack, which therefore contains only 36 cards. Two cards are drawn at random from the pack. The probability that both cards are red is \(\frac{1}{3}\).

1. Show that \(r\), the number of red cards in the pack, satisfies the equation $$r(r - 1) = 420.$$ [4 marks]
2. Hence or otherwise find the value of \(r\). [3 marks]
3. Find the probability that, when three cards are drawn at random from the pack,
   1. at least two are red, [6 marks]
   2. the first one is red given that at least two are red. [4 marks]

The events \(A\) and \(B\) are such that P\((A \cap B) = 0.24\), P\((A \cup B) = 0.88\) and P\((B) = 0.52\).

1. Find P\((A)\). [3 marks]
2. Determine, with reasons, whether \(A\) and \(B\) are
   1. mutually exclusive,
   2. independent.
   [4 marks]
3. Find P\((B | A)\). [2 marks]
4. Find P\((A' | B')\). [3 marks]

Of the cars that are taken to a certain garage for an M.O.T. test, 87% pass. However, 2% of these have faults for which they should have been failed. 5% of the cars which fail are in fact roadworthy and should have passed. Using a tree diagram, or otherwise, calculate the probabilities that a car chosen at random

1. should have passed the test, regardless of whether it actually did or not, [4 marks]
2. failed the test, given that it should have passed. [3 marks]

The garage is told to improve its procedures. When it is inspected again a year later, it is found that the pass rate is still 87% overall and 2% of the cars passed have faults as before, but now 0.3% of the cars which should have passed are failed and \(x\)% of the cars which are failed should have passed.

1. Find the value of \(x\). [8 marks]

\(A\), \(B\) and \(C\) are three events such that \(\text{P}(A) = x\), \(\text{P}(B) = y\) and \(\text{P}(C) = x + y\). It is known that \(\text{P}(A \cup B) = 0.6\) and \(\text{P}(B \mid A) = 0.2\).

1. Show that \(4x + 5y = 3\). [2 marks]

It is also known that \(B\) and \(C\) are mutually exclusive and that \(\text{P}(B \cup C) = 0.9\)

1. Obtain another equation in \(x\) and \(y\) and hence find the values of \(x\) and \(y\). [4 marks]
2. Deduce whether or not \(A\) and \(B\) are independent events. [2 marks]

Among the families with two children in a large city, the probability that the elder child is a boy is \(\frac{5}{12}\) and the probability that the younger child is a boy is \(\frac{9}{16}\). The probability that the younger child is a girl, given that the elder child is a girl, is \(\frac{1}{4}\). One of the families is chosen at random. Using a tree diagram, or otherwise,

1. show that the probability that both children are boys is \(\frac{1}{8}\). [5 marks]

Find the probability that

1. one child is a boy and the other is a girl, [3 marks]
2. one child is a boy given that the other is a girl. [3 marks]

If three of the families are chosen at random,

1. find the probability that exactly two of the families have two boys. [3 marks]
2. State an assumption that you have made in answering part (d). [1 mark]

The students in a large Sixth Form can choose to do exactly one of Community Service, Games or Private Study on Wednesday afternoons. The probabilities that a randomly chosen student does Games and Private Study are \(\frac{3}{8}\) and \(\frac{1}{5}\) respectively. It may be assumed that the number of students is large enough for these probabilities to be treated as constant.

1. Find the probability that a randomly chosen student does Community Service. [2 marks]
2. If two students are chosen at random, find the probability that they both do the same activity. [3 marks]
3. If three students are chosen at random, find the probability that exactly one of them does Games. [3 marks]

Two-fifths of the students are girls, and a quarter of these girls do Private Study.

1. Find the probability that a randomly chosen student who does Private Study is a boy. [5 marks]

The table shows the numbers of male and female members of a vintage car club who own either a Jaguar or a Bentley. No member owns both makes of car.

	Male	Female
Jaguar	25	15
Bentley	12	8

One member is chosen at random from these 60 members.

1. Given that this member is male, find the probability that he owns a Jaguar. [2]

Now two members are chosen at random from the 60 members. They are chosen one at a time, without replacement.

1. Given that the first one of these members is female, find the probability that both own Jaguars. [4]

Andy can walk to work, travel by bike or travel by bus. The tree diagram shows the probabilities of any day being dry or wet and the corresponding probabilities for each of Andy's methods of travel. \includegraphics{figure_5} A day is selected at random. Find the probability that

1. the weather is wet and Andy travels by bus, [2]
2. Andy walks or travels by bike, [3]
3. the weather is dry given that Andy walks or travels by bike. [3]

Each weekday, Marta travels to school by bus. Sometimes she arrives late. • \(L\) is the event that Marta arrives late. • \(R\) is the event that it is raining. You are given that \(\mathrm{P}(L) = 0.15\), \(\mathrm{P}(R) = 0.22\) and \(\mathrm{P}(L \mid R) = 0.45\).

1. Use this information to show that the events \(L\) and \(R\) are not independent. [1]
2. Find \(\mathrm{P}(L \cap R)\). [2]
3. Draw a Venn diagram showing the events \(L\) and \(R\), and fill in the probability corresponding to each of the four regions of your diagram. [3]

The events \(A\) and \(B\) are such that $$\text{P}(A) = \frac{5}{16}, \text{P}(B) = \frac{1}{2} \text{ and P}(A|B) = \frac{1}{4}$$ Find

1. P\((A \cap B)\). [2]
2. P\((B'|A)\). [3]
3. P\((A' \cup B)\). [2]
4. Determine, with a reason, whether or not the events \(A\) and \(B\) are independent. [3]

The events \(A\) and \(B\) are such that $$\text{P}(A) = \frac{7}{12}, \quad \text{P}(A \cap B) = \frac{1}{4} \quad \text{and} \quad \text{P}(A|B) = \frac{2}{3}.$$ Find

1. P\((B)\), [3 marks]
2. P\((A \cup B)\), [3 marks]
3. P\((B|A')\). [3 marks]

The events \(A\) and \(B\) are independent and such that $$\text{P}(A) = 2\text{P}(B) \text{ and } \text{P}(A \cap B) = \frac{1}{8}.$$

1. Show that \(\text{P}(B) = \frac{1}{4}\). [5 marks]
2. Find \(\text{P}(A \cup B)\). [3 marks]
3. Find \(\text{P}(A | B')\). [2 marks]

In a food survey, a large number of people are asked whether they like tomato soup, mushroom soup, both or neither. One of these people is selected at random. • \(T\) is the event that this person likes tomato soup. • \(M\) is the event that this person likes mushroom soup. You are given that \(\text{P}(T) = 0.55\), \(\text{P}(M) = 0.33\) and \(\text{P}(T|M) = 0.80\).

1. Use this information to show that the events \(T\) and \(M\) are not independent. [1]
2. Find \(\text{P}(T \cap M)\). [2]
3. Draw a Venn diagram showing the events \(T\) and \(M\), and fill in the probability corresponding to each of the four regions of your diagram. [3]