2.04a Discrete probability distributions

7 A fair die has one face numbered 1, one face numbered 3, two faces numbered 5 and two faces numbered 6 .

1. Find the probability of obtaining at least 7 odd numbers in 8 throws of the die. The die is thrown twice. Let \(X\) be the sum of the two scores. The following table shows the possible values of \(X\). \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Second throw}

   		1	3	5	5	6	6
   \cline { 2 - 8 }	1	2	4	6	6	7	7
   	3	4	6	8	8	9	9
   First	5	6	8	10	10	11	11
   throw	5	6	8	10	10	11	11
   	6	7	9	11	11	12	12
   	6	7	9	11	11	12	12

   \end{table}
2. Draw up a table showing the probability distribution of \(X\).
3. Calculate \(\mathrm { E } ( X )\).
4. Find the probability that \(X\) is greater than \(\mathrm { E } ( X )\).

2 Two unbiased tetrahedral dice each have four faces numbered \(1,2,3\) and 4. The two dice are thrown together and the sum of the numbers on the faces on which they land is noted. Find the expected number of occasions on which this sum is 7 or more when the dice are thrown together 200 times.

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 .

1 The discrete random variable \(X\) takes the values 1, 4, 5, 7 and 9 only. The probability distribution of \(X\) is shown in the table. Find \(p\).

6 A fair tetrahedral die has four triangular faces, numbered \(1,2,3\) and 4 . The score when this die is thrown is the number on the face that the die lands on. This die is thrown three times. The random variable \(X\) is the sum of the three scores.

1. Show that \(\mathrm { P } ( X = 9 ) = \frac { 10 } { 64 }\).
2. Copy and complete the probability distribution table for \(X\).

   \(x\)	3	4	5	6	7	8	9	10	11	12
   \(\mathrm { P } ( X = x )\)	\(\frac { 1 } { 64 }\)	\(\frac { 3 } { 64 }\)			\(\frac { 12 } { 64 }\)

3. Event \(R\) is 'the sum of the three scores is 9 '. Event \(S\) is 'the product of the three scores is 16 '. Determine whether events \(R\) and \(S\) are independent, showing your working.

2 The discrete random variable \(X\) has the following probability distribution. Given that \(\mathrm { E } ( X ) = 2.3\) and \(\operatorname { Var } ( X ) = 3.01\), find the values of \(p , q\) and \(r\).

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.

7 Rory has 10 cards. Four of the cards have a 3 printed on them and six of the cards have a 4 printed on them. He takes three cards at random, without replacement, and adds up the numbers on the cards.

1. Show that P (the sum of the numbers on the three cards is \(11 ) = \frac { 1 } { 2 }\).
2. Draw up a probability distribution table for the sum of the numbers on the three cards. Event \(R\) is 'the sum of the numbers on the three cards is 11 '. Event \(S\) is 'the number on the first card taken is a \(3 ^ { \prime }\).
3. Determine whether events \(R\) and \(S\) are independent. Justify your answer.
4. Determine whether events \(R\) and \(S\) are exclusive. Justify your answer.

7 Dayo chooses two digits at random, without replacement, from the 9-digit number 113333555.

1. Find the probability that the two digits chosen are equal.
2. Find the probability that one digit is a 5 and one digit is not a 5 .
3. Find the probability that the first digit Dayo chose was a 5, given that the second digit he chose is not a 5 .
4. The random variable \(X\) is the number of 5s that Dayo chooses. Draw up a table to show the probability distribution of \(X\).

2 The number of phone calls, \(X\), received per day by Sarah has the following probability distribution.

1. Find the value of \(k\).
2. Find the mode of \(X\).
3. Find the probability that the number of phone calls received by Sarah on any particular day is more than the mean number of phone calls received per day.

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.

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.

5 A fair red spinner has four sides, numbered 1, 2, 3, 3. A fair blue spinner has three sides, numbered \(- 1,0,2\). When a spinner is spun, the score is the number on the side on which it lands. The spinners are spun at the same time. The random variable \(X\) denotes the score on the red spinner minus the score on the blue spinner.

1. Draw up the probability distribution table for \(X\).
2. Find \(\operatorname { Var } ( X )\).

6 A box contains 3 red balls and 5 white balls. One ball is chosen at random from the box and is not returned to the box. A second ball is now chosen at random from the box.

1. Find the probability that both balls chosen are red.
2. Show that the probability that the balls chosen are of different colours is \(\frac { 15 } { 28 }\).
3. Given that the second ball chosen is red, find the probability that the first ball chosen is red.
   The random variable \(X\) denotes the number of red balls chosen.
4. Draw up the probability distribution table for \(X\).
5. Find \(\operatorname { Var } ( X )\).

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.

7 The faces of a biased die are numbered \(1,2,3,4,5\) and 6 . The probabilities of throwing odd numbers are all the same. The probabilities of throwing even numbers are all the same. The probability of throwing an odd number is twice the probability of throwing an even number.

1. Find the probability of throwing a 3 . \includegraphics[max width=\textwidth, alt={}, center]{34ae4f06-d485-4138-82d8-902b70f08995-10_51_1563_495_331}
2. The die is thrown three times. Find the probability of throwing two 5 s and one 4 .
3. The die is thrown 100 times. Use an approximation to find the probability that an even number is thrown at most 37 times.

2 In a probability distribution the random variable \(X\) takes the value \(x\) with probability \(k x\), where \(x\) takes values \(1,2,3,4,5\) only.

1. Draw up a probability distribution table for \(X\), in terms of \(k\), and find the value of \(k\).
2. Find \(\mathrm { E } ( X )\).

3 A team of 4 is to be randomly chosen from 3 boys and 5 girls. The random variable \(X\) is the number of girls in the team.

1. Draw up a probability distribution table for \(X\).
2. Given that \(\mathrm { E } ( X ) = \frac { 5 } { 2 }\), calculate \(\operatorname { Var } ( X )\).

4 A fair six-sided die is rolled twice. The random variable \(X\) represents the higher of the two scores. The probability distribution of \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k ( 2 r - 1 ) \text { for } r = 1,2,3,4,5,6 .$$

1. Copy and complete the following probability table and hence find the exact value of \(k\), giving your answer as a fraction in its simplest form.

   \(r\)	1	2	3	4	5	6
   \(\mathrm { P } ( X = r )\)	\(k\)					\(11 k\)

2. Find the mean of \(X\). A fair six-sided die is rolled three times.
3. Find the probability that the total score is 16 .

4 A company is searching for oil reserves. The company has purchased the rights to make test drillings at four sites. It investigates these sites one at a time but, if oil is found, it does not proceed to any further sites. At each site, there is probability 0.2 of finding oil, independently of all other sites. The random variable \(X\) represents the number of sites investigated. The probability distribution of \(X\) is shown below.

1. Find the expectation and variance of \(X\).
2. It costs \(\pounds 45000\) to investigate each site. Find the expected total cost of the investigation.
3. Draw a suitable diagram to illustrate the distribution of \(X\).

3 Jeremy is a computing consultant who sometimes works at home. The number, \(X\), of days that Jeremy works at home in any given week is modelled by the probability distribution $$\mathrm { P } ( X = r ) = \frac { 1 } { 40 } r ( r + 1 ) \quad \text { for } r = 1,2,3,4 .$$

1. Verify that \(\mathrm { P } ( X = 4 ) = \frac { 1 } { 2 }\).
2. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
3. Jeremy works for 45 weeks each year. Find the expected number of weeks during which he works at home for exactly 2 days.

3 The score, \(X\), obtained on a given throw of a biased, four-faced die is given by the probability distribution $$\mathrm { P } ( X = r ) = k r ( 8 - r ) \text { for } r = 1,2,3,4 .$$

1. Show that \(k = \frac { 1 } { 50 }\).
2. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

6 In a phone-in competition run by a local radio station, listeners are given the names of 7 local personalities and are told that 4 of them are in the studio. Competitors phone in and guess which 4 are in the studio.

1. Show that the probability that a randomly selected competitor guesses all 4 correctly is \(\frac { 1 } { 35 }\). Let \(X\) represent the number of correct guesses made by a randomly selected competitor. The probability distribution of \(X\) is shown in the table.

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	0	\(\frac { 4 } { 35 }\)	\(\frac { 18 } { 35 }\)	\(\frac { 12 } { 35 }\)	\(\frac { 1 } { 35 }\)

2. Find the expectation and variance of \(X\).

3 In a game of darts, a player throws three darts. Let \(X\) represent the number of darts which hit the bull's-eye. The probability distribution of \(X\) is shown in the table.

1. (A) Show that \(p + q = 0.15\).
   (B) Given that the expectation of \(X\) is 0.67 , show that \(2 p + 3 q = 0.32\).
   (C) Find the values of \(p\) and \(q\).
2. Find the variance of \(X\).

2 In a game of darts, a player throws three darts. Let \(X\) represent the number of darts which hit the bull's-eye. The probability distribution of \(X\) is shown in the table.

1. (A) Show that \(p + q = 0.15\).
   (B) Given that the expectation of \(X\) is 0.67 , show that \(2 p + 3 q = 0.32\).
   (C) Find the values of \(p\) and \(q\).
2. Find the variance of \(X\).