Sequential trials until success

2 Gohan throws a fair tetrahedral die with faces numbered \(1,2,3,4\). If she throws an even number then her score is the number thrown. If she throws an odd number then she throws again and her score is the sum of both numbers thrown. Let the random variable \(X\) denote Gohan's score.

1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 5 } { 16 }\).
2. The table below shows the probability distribution of \(X\).

   \(x\)	2	3	4	5	6	7
   \(\mathrm { P } ( X = x )\)	\(\frac { 5 } { 16 }\)	\(\frac { 1 } { 16 }\)	\(\frac { 3 } { 8 }\)	\(\frac { 1 } { 8 }\)	\(\frac { 1 } { 16 }\)	\(\frac { 1 } { 16 }\)

   Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

3 A box contains 6 identical-sized discs, of which 4 are blue and 2 are red. Discs are taken at random from the box in turn and not replaced. Let \(X\) be the number of discs taken, up to and including the first blue one.

1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 1 } { 15 }\).
2. Draw up the probability distribution table for \(X\).

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 \(X\).

2 A couple plan to have at least one child of each sex, after which they will have no more children. However, if they have four children of one sex, they will have no more children. You should assume that each child is equally likely to be of either sex, and that the sexes of the children are independent. The random variable \(X\) represents the total number of girls the couple have.

1. Show that \(\mathrm { P } ( X = 1 ) = \frac { 11 } { 16 }\). The table shows the probability distribution of \(X\).

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	\(\frac { 1 } { 16 }\)	\(\frac { 11 } { 16 }\)	\(\frac { 1 } { 8 }\)	\(\frac { 1 } { 16 }\)	\(\frac { 1 } { 16 }\)

2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

7 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 \(X\).

5. A biased tetrahedral die has faces numbered \(0,1,2\) and 3 . The die is rolled and the number face down on the die, \(X\), is recorded. The probability distribution of \(X\) is If \(X = 3\) then the final score is 3
If \(X \neq 3\) then the die is rolled again and the final score is the sum of the two numbers. The random variable \(T\) is the final score.

1. Find \(\mathrm { P } ( T = 2 )\)
2. Find \(\mathrm { P } ( T = 3 )\)
3. Given that the die is rolled twice, find the probability that the final score is 3

1. A biased tetrahedral die has faces numbered \(0,1,2\) and 3 . The die is rolled and the number face down on the die, \(X\), is recorded. The probability distribution of \(X\) is

If \(X = 3\) then the final score is 3
If \(X \neq 3\) then the die is rolled again and the final score is the sum of the two numbers.
The random variable \(T\) is the final score.

1. Find \(\mathrm { P } ( T = 2 )\)
2. Find \(\mathrm { P } ( T = 3 )\)
3. Given that the die is rolled twice, find the probability that the final score is 3 \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
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5

1. Joshua plays a game in which he repeatedly tosses an unbiased coin. His game concludes when he obtains either a head or 5 tails in succession. The random variable \(N\) denotes the number of tosses of his coin required to conclude a game. By completing Table 3 below, calculate \(\mathrm { E } ( N )\).
2. Joshua's sister, Ruth, plays a separate game in which she repeatedly tosses a coin that is biased in such a way that the probability of a head in a single toss of her coin is \(\frac { 1 } { 4 }\). Her game also concludes when she obtains either a head or 5 tails in succession. The random variable \(M\) denotes the number of tosses of her coin required to conclude her game. Complete Table 4 below.
3. 1. Joshua and Ruth play their games simultaneously. Calculate the probability that Joshua and Ruth will conclude their games in an equal number of tosses of their coins.
   2. Joshua and Ruth play their games simultaneously on 3 occasions. Calculate the probability that, on at least 2 of these occasions, their games will be concluded in an equal number of tosses of their coins. Give your answer to three decimal places.
      (4 marks) \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Table 3}

      \(\boldsymbol { n }\)	1	2	3	4	5
      \(\mathbf { P } ( \boldsymbol { N } = \boldsymbol { n } )\)			\(\frac { 1 } { 8 }\)		\(\frac { 1 } { 16 }\)

      \end{table} \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Table 4}

      \(\boldsymbol { m }\)	1	2	3	4	5
      \(\mathbf { P } ( \boldsymbol { M } = \boldsymbol { m } )\)	\(\frac { 1 } { 4 }\)	\(\frac { 3 } { 16 }\)

      \end{table}

1 In a promotion for a new type of cereal, a toy dinosaur is included in each pack. There are three different types of dinosaur to collect. They are distributed, with equal probability, randomly and independently in the packs. Sam is trying to collect all three of the dinosaurs.

1. Find the probability that Sam has to open only 3 packs in order to collect all three dinosaurs. Sam continues to open packs until she has collected all three dinosaurs, but once she has opened 6 packs she gives up even if she has not found all three. The random variable \(X\) represents the number of packs which Sam opens.
2. Complete the table below, using the copy in the Printed Answer Booklet, to show the probability distribution of \(X\).

   \(r\)	3	4	5	6
   \(\mathrm { P } ( X = r )\)		\(\frac { 2 } { 9 }\)	\(\frac { 14 } { 81 }\)

   \section*{(iii) In this question you must show detailed reasoning.} Find
   • \(\mathrm { E } ( X )\) and
   • \(\operatorname { Var } ( X )\).

Four children are playing a game in order to win a calculator. They take turns, starting with Alex, followed by Ben, then Caroline, then Danielle, rolling a fair six-sided dice until someone obtains a 6. This player then wins a calculator.

1. Find the probability that
   1. Danielle wins the calculator on her first turn, [1]
   2. Ben wins the calculator on his first or second turn, [3]
   3. Caroline rolls the dice exactly twice. [3]
2. Show that the probability that Alex wins the calculator is \(\frac{216}{671}\). [3]