First success on specific trial

4 Three fair six-sided dice, each with faces marked \(1,2,3,4,5,6\), are thrown at the same time, repeatedly. For a single throw of the three dice, the score is the sum of the numbers on the top faces.

1. Find the probability that the score is 4 on a single throw of the three dice.
2. Find the probability that a score of 18 is obtained for the first time on the 5th throw of the three dice.

4 Ramesh throws an ordinary fair 6-sided die.

1. Find the probability that he obtains a 4 for the first time on his 8th throw.
2. Find the probability that it takes no more than 5 throws for Ramesh to obtain a 4 .
   Ramesh now repeatedly throws two ordinary fair 6-sided dice at the same time. Each time he adds the two numbers that he obtains.
3. For 10 randomly chosen throws of the two dice, find the probability that Ramesh obtains a total of less than 4 on at least three throws.

1 A fair spinner with 5 sides numbered 1,2,3,4,5 is spun repeatedly. The score on each spin is the number on the side on which the spinner lands.

1. Find the probability that a score of 3 is obtained for the first time on the 8th spin.
2. Find the probability that fewer than 6 spins are required to obtain a score of 3 for the first time.

3 One plastic robot is given away free inside each packet of a certain brand of biscuits. There are four colours of plastic robot (red, yellow, blue and green) and each colour is equally likely to occur. Nick buys some packets of these biscuits. Find the probability that

1. he gets a green robot on opening his first packet,
2. he gets his first green robot on opening his fifth packet. Nick's friend Amos is also collecting robots.
3. Find the probability that the first four packets Amos opens all contain different coloured robots.

8 The proportion of people who watch West Street on television is \(30 \%\). A market researcher interviews people at random in order to contact viewers of West Street. Each day she has to contact a certain number of viewers of West Street.

1. Near the end of one day she finds that she needs to contact just one more viewer of West Street. Find the probability that the number of further interviews required is
   (a) 4 ,

5 Malik is playing a game in which he has to throw a 6 on a fair six-sided die to start the game. Find the probability that

1. Malik throws a 6 for the first time on his third attempt,
2. Malik needs at most ten attempts to throw a 6.

4 It is known that \(8 \%\) of the population of a large city use a particular web browser. A researcher wishes to interview some people from the city who use this browser. He selects people at random, one at a time.

1. Find the probability that the first person that he finds who uses this browser is
   (A) the third person selected,
   (B) the second or third person selected.
2. Find the probability that at least one of the first 20 people selected uses this browser.

6 The probability that a particular type of light bulb is defective is 0.01 . A large number of these bulbs are tested, one by one. Assuming independence, find the probability that the tenth bulb tested is the first to be found defective. The first defective bulb is the \(N\) th to be tested. Write down the value of \(\mathrm { E } ( N )\). Find the least value of \(n\) such that \(\mathrm { P } ( N \leqslant n )\) is greater than 0.9 .

4. A customer wishes to withdraw money from a cash machine. To do this it is necessary to type a PIN number into the machine. The customer is unsure of this number. If the wrong number is typed in, the customer can try again up to a maximum of four attempts in total. Attempts to type in the correct number are independent and the probability of success at each attempt is 0.6 .

1. Show that the probability that the customer types in the correct number at the third attempt is 0.096 .
   (2 marks)
   The random variable \(A\) represents the number of attempts made to type in the correct PIN number, regardless of whether or not the attempt is successful.
2. Find the probability distribution of \(A\).
3. Calculate the probability that the customer types in the correct number in four or fewer attempts.
4. Calculate \(\mathrm { E } ( A )\) and \(\operatorname { Var } ( A )\).
5. Find \(\mathrm { F } ( 1 + \mathrm { E } ( A ) )\).

4 The probability that an expert darts player hits the bullseye on any throw is 0.12 , independently of any other throw. The player throws darts at the bullseye until she hits it.

1. Find the probability that the player has to throw exactly six darts.
2. Find the probability that the player has to throw more than six darts.
3. (A) Find the mean number of darts that the player has to throw.
   (B) Find the variance of the number of darts that the player has to throw. The player continues to throw more darts at the bullseye after she has hit it for the first time.
4. Find the probability that the player hits the bullseye at least twice in the first ten throws.
5. Find the probability that the player hits the bullseye for the second time on the tenth throw.

2 A football player is practising taking penalties. On each attempt the player has a \(70 \%\) chance of scoring a goal. The random variable \(X\) represents the number of attempts that it takes for the player to score a goal.

1. Determine \(\mathrm { P } ( X = 4 )\).
2. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   • Determine the probability that the player needs exactly 4 attempts to score 2 goals.
   • The player has \(n\) attempts to score a goal.
     1. Determine the least value of \(n\) for which the probability that the player first scores a goal on the \(n\)th attempt is less than 0.001 .
     2. Determine the least value of \(n\) for which the probability that the player scores at least one goal in \(n\) attempts is at least 0.999.

5 In a recent report, it was stated that \(40 \%\) of working people have a degree. For the whole of this question, you should assume that this is true. A researcher wishes to interview a working person who has a degree. He asks working people at random whether they have a degree and counts the number of people he has to ask until he finds one with a degree.

1. Find the probability that he has to ask 5 people.
2. Find the mean number of people the researcher has to ask. Subsequently, the researcher decides to take a random sample from the population of working people.
3. A random sample of 5 working people is chosen. What is the probability that at least one of them has a degree?
4. How large a random sample of working people would the researcher need to take to ensure that the probability that at least one person has a degree is 0.99 or more?

2 A market researcher wants to interview people who watched a particular television programme. Audience research data used by the broadcaster indicates that \(12 \%\) of the adult population watched this programme. This figure is used to model the situation.
The researcher asks people in a shopping centre, one at a time, if they watched the programme. You should assume that these people form a random sample of the adult population.

1. Find the probability that the fifth person the researcher asks is the first to have watched the programme.
2. Find the probability that the researcher has to ask at least 10 people in order to find one who watched the programme.
3. Find the probability that the twentieth person the researcher asks is the third to have watched the programme.
4. Find how many people the researcher would have to ask to ensure that there is a probability of at least 0.95 that at least one of them watched the programme.

1 A fair spinner has ten sectors, labelled \(1,2 , \ldots , 10\). In order to start a game, Kofi has to obtain an 8,9 or 10 on the spinner.

1. Find the probability that Kofi starts the game on the third spin.
2. Find the probability that Kofi takes at least 5 spins to start the game.
3. Determine the probability that the number of spins required to start the game is within 1 standard deviation of its mean.

1 A darts player is trying to hit the bullseye on a dart board. On each throw the probability that she hits it is 0.05 , independently of any other throw.

1. Find the probability that she hits the bullseye for the first time on her 10th throw.
2. Find the probability that she does not hit the bullseye in her first 10 throws.
3. Write down the expected number of throws which it takes her to hit the bullseye for the first time.
4. Write down the expected number of throws which it takes her to hit the bullseye for the first time.

7 Amir is trying to thread a needle. On each attempt the probability that he is successful is 0.3 , independently of any other attempt. The random variable \(X\) represents the number of attempts that he takes to thread the needle.

1. Find \(\mathrm { P } ( X = 5 )\).
2. During the course of a day, Amir has to thread 6 needles. Determine the probability that it takes him more than 3 attempts to be successful for at least 4 of the 6 needles.
3. Amaya is also trying to thread a needle. On each attempt the probability that she is successful is \(p\), independently of any other attempt. The probability that Amaya takes 2 attempts to thread a particular needle is \(\frac { 28 } { 121 }\). Determine the possible values of \(p\).

3. A football player is practising taking penalties. On each attempt the player has a \(70 \%\) chance of scoring a goal. The random variable \(X\) represents the number of attempts that it takes for the player to score a goal.

1. Determine \(\mathrm { P } ( X = 4 )\).
2. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   • Determine the probability that the player needs exactly 4 attempts to score 2 goals.
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4 A fair six-sided dice is rolled repeatedly. Find the probability of the following events.

1. A five occurs for the first time on the fourth roll.
2. A five occurs at least once in the first four rolls.
3. A five occurs for the second time on the third roll.
4. At least two fives occur in the first three rolls. The dice is rolled repeatedly until a five occurs for the second time.
5. Find the expected number of rolls required for two fives to occur. Justify your answer.

4. A market researcher wants to interview people who watched a particular television programme. Audience research data used by the broadcaster indicates that \(12 \%\) of the adult population watched this programme. This figure is used to model the situation.
The researcher asks people in a shopping centre, one at a time, if they watched the programme. You should assume that these people form a random sample of the adult population.

1. Find the probability that the fifth person the researcher asks is the first to have watched the programme.
2. Find the probability that the researcher has to ask at least 10 people in order to find one who watched the programme.
3. Find the probability that the twentieth person the researcher asks is the third to have watched the programme.
4. Find how many people the researcher would have to ask to ensure that there is a probability of at least 0.95 that at least one of them watched the programme.
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