5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1)

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. [2] The first defective bulb is the \(N\)th to be tested. Write down the value of E(\(N\)). [1] Find the least value of \(n\) such that P(\(N \leqslant n\)) is greater than \(0.9\). [3]

A fair die is thrown repeatedly until a 6 is obtained.

1. Find the probability that obtaining a 6 takes no more than four throws. [2]
2. Find the least integer \(N\) such that the probability of obtaining a 6 before the \(N\)th throw is more than 0.95. [3]

A fair die is thrown repeatedly until a 6 is obtained.

1. Find the probability that obtaining a 6 takes no more than four throws. [2]
2. Find the least integer \(N\) such that the probability of obtaining a 6 before the \(N\)th throw is more than 0.95. [3]

A fair six-sided die is thrown until a 3 or a 4 is obtained. The number of throws taken is denoted by the random variable \(X\).

1. State the mean value of \(X\). [1]
2. Find the probability that obtaining a 3 or a 4 takes exactly 6 throws. [1]
3. Find the probability that obtaining a 3 or a 4 takes more than 4 throws. [2]
4. Find the greatest integer \(n\) such that the probability of obtaining a 3 or a 4 in fewer than \(n\) throws is less than 0.95. [3]

An archer shoots at a target. It may be assumed that each shot is independent of all other shots and that, on average, she hits the bull's-eye with 3 shots in 20. Find the probability that she requires at least 6 shots to hit the bull's-eye. [3] When she hits the bull's-eye for the third time her total number of shots is \(Y\). Show that $$\mathrm{P}(Y = r) = \frac{1}{2}(r - 1)(r - 2)\left(\frac{3}{20}\right)^3\left(\frac{17}{20}\right)^{r-3}.$$ [3] Simplify \(\frac{\mathrm{P}(Y = r + 1)}{\mathrm{P}(Y = r)}\), and hence find the set of values of \(r\) for which \(\mathrm{P}(Y = r + 1) < \mathrm{P}(Y = r)\). Deduce the most probable value of \(Y\). [4]

The discrete random variable \(X\) has a geometric distribution with mean 4. Find

1. P\((X = 5)\), [3]
2. P\((X > 5)\), [2]
3. the least integer \(N\) such that P\((X \leqslant N) > 0.9995\). [2]

A biased coin is tossed repeatedly until a head is obtained. The random variable \(X\) denotes the number of tosses required for a head to be obtained. The mean of \(X\) is equal to twice the variance of \(X\). Show that the probability that a head is obtained when the coin is tossed once is \(\frac{2}{3}\). [2] Find

1. P(\(X = 4\)), [1]
2. P(\(X > 4\)), [2]
3. the least integer \(N\) such that P(\(X \leq N\)) \(> 0.999\). [3]

Lan starts a new job on Monday. He will catch the bus to work every day from Monday to Friday inclusive. The probability that he will get a seat on the bus has the constant value \(p\). The random variable \(X\) denotes the number of days that Lan will catch the bus until he is able to get a seat. The probability that Lan will not get a seat on the Monday, Tuesday, Wednesday or Thursday of his first week is 0.4096.

1. Show that \(p = 0.2\). [2]
2. Find the probability that Lan first gets a seat on Monday of the second week in his new job. [2]
3. Find the least integer \(N\) such that \(\text{P}(X \leqslant N) > 0.9\), and identify the day and the week that corresponds to this value of \(N\). [4]

1. Write down the conditions under which the binomial distribution may be a suitable model to use in statistical work. [4]

A six-sided die is biased. When the die is thrown the number 5 is twice as likely to appear as any other number. All the other faces are equally likely to appear. The die is thrown repeatedly. Find the probability that

1. the first 5 will occur on the sixth throw, [8]
2. in the first eight throws there will be exactly three 5s.

1. Write down the conditions under which the binomial distribution may be a suitable model to use in statistical work. [4]

A six-sided die is biased. When the die is thrown the number 5 is twice as likely to appear as any other number. All the other faces are equally likely to appear. The die is thrown repeatedly. Find the probability that

1. 1. the first 5 will occur on the sixth throw,
   2. in the first eight throws there will be exactly three 5s.
   [8]

Andy makes repeated attempts to thread a needle. The number of attempts up to and including his first success is denoted by \(X\).

1. State two conditions necessary for \(X\) to have a geometric distribution. [2]
2. Assuming that \(X\) has the distribution Geo(0.3), find
   1. P\((X = 5)\), [2]
   2. P\((X > 5)\). [3]
3. Suggest a reason why one of the conditions you have given in part (i) might not be satisfied in this context. [2]

\(R\) and \(S\) are independent random variables each having the distribution Geo\((p)\).

1. Find P\((R = 1\) and \(S = 1)\) in terms of \(p\). [1]
2. Show that P\((R = 3\) and \(S = 3) = p^2q^4\), where \(q = 1 - p\). [1]
3. Use the formula for the sum to infinity of a geometric series to show that $$\text{P}(R = S) = \frac{p}{2-p}.$$ [5]

Sandra makes repeated, independent attempts to hit a target. On each attempt, the probability that she succeeds is 0.1.

1. Find the probability that
   1. the first time she succeeds is on her 5th attempt, [2]
   2. the first time she succeeds is after her 5th attempt, [2]
   3. the second time she succeeds is before her 4th attempt. [4]
   Jill also makes repeated attempts to hit the target. Each attempt of either Jill or Sandra is independent. Each time that Jill attempts to hit the target, the probability that she succeeds is 0.2. Sandra and Jill take turns attempting to hit the target, with Sandra going first.
2. Find the probability that the first person to hit the target is Sandra, on her
   1. 2nd attempt, [2]
   2. 10th attempt. [3]

30% of people own a Talk-2 phone. People are selected at random, one at a time, and asked whether they own a Talk-2 phone. The number of people questioned, up to and including the first person who owns a Talk-2 phone, is denoted by \(X\). Find

1. P(\(X = 4\)), [3]
2. P(\(X > 4\)), [2]
3. P(\(X < 6\)). [3]

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
   1. 4, [3]
   2. less than 4. [3]
2. Near the end of another day she finds that she needs to contact just two more viewers of West Street. Find the probability that the number of further interviews required is
   1. 5, [4]
   2. more than 5. [2]

In a factory, an inspector checks a random sample of 30 mugs from a large batch and notes the number, \(X\), which are defective. He then deals with the batch as follows. • If \(X < 2\), the batch is accepted. • If \(X > 2\), the batch is rejected. • If \(X = 2\), the inspector selects another random sample of only 15 mugs from the batch. If this second sample contains 1 or more defective mugs, the batch is rejected. Otherwise the batch is accepted. It is given that 5\% of mugs are defective.

1. 1. Find the probability that the batch is rejected after just the first sample is checked. [3]
   2. Show that the probability that the batch is rejected is 0.327, correct to 3 significant figures. [5]
2. Batches are checked one after another. Find the probability that the first batch to be rejected is either the 4th or the 5th batch that is checked. [3]

A game is played with a token on a board with a grid printed on it. The token starts at the point \((0, 0)\) and moves in steps. Each step is either 1 unit in the positive \(x\)-direction with probability 0.8, or 1 unit in the positive \(y\)-direction with probability 0.2. The token stops when it reaches a point with a \(y\)-coordinate of 1. It is given that the token stops at \((X, 1)\).

1. 1. Find the probability that \(X = 10\). [2]
   2. Find the probability that \(X < 10\). [3]
2. Find the expected number of steps taken by the token. [2]
3. Hence, write down the value of E\((X)\). [1]

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
   1. the third person selected, [3]
   2. the second or third person selected. [2]
2. Find the probability that at least one of the first 20 people selected uses this browser. [3]

A biased spinner has five sides, numbered 1 to 5. Elmer spins the spinner repeatedly and counts the number of spins, \(X\), up to and including the first time that the number 2 appears. He carries out this experiment 100 times and records the frequency \(f\) with which each value of \(X\) is obtained. His results are shown in Table 1, together with the values of \(xf\).

\(x\)	1	2	3	4	5	6	\(\geqslant 7\)	Total
Frequency \(f\)	20	15	9	13	10	10	23	100
\(xf\)	20	30	27	52	50	60	161	400

Table 1

1. State an appropriate distribution with which to model \(X\), determining the value(s) of any parameter(s). [3]

Elmer carries out a goodness-of-fit test, at the 5\% level, for the distribution in part (a). Table 2 gives some of his calculations, in which numbers that are not exact have been rounded to 3 decimal places.

\(x\)	1	2	3	4	5	6	\(\geqslant 7\)
Observed frequency \(O\)	20	15	9	13	10	10	23
Expected frequency \(E\)	25	18.75	14.063	10.547	7.910	5.933	17.798
\((O - E)^2/E\)	1	0.75	1.823	0.571	0.552	2.789	1.520

Table 2

1. Show how the expected frequency corresponding to \(x \geqslant 7\) was obtained. [2]
2. Carry out the test. [5]

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 \(10\)th throw. [2]
2. Find the probability that she does not hit the bullseye in her first \(10\) throws. [1]
3. Write down the expected number of throws which it takes her to hit the bullseye for the first time. [1]

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]

Every time a spinner is spun, the probability that it shows the number 4 is 0.2, independently of all other spins.

1. A pupil spins the spinner repeatedly until it shows the number 4. Find the mean of the number of spins required. [2]
2. Calculate the probability that the number of spins required is between 3 and 10 inclusive. [2]
3. Each pupil in a class of 30 spins the spinner until it shows the number 4. Out of the 30 pupils, the number of pupils who require at least 10 spins is denoted by \(X\). Determine the variance of \(X\). [4]

A spinner can land on red or blue. When the spinner is spun, there is a probability of \(\frac{1}{3}\) that it lands on blue. The spinner is spun repeatedly. The random variable \(B\) represents the number of the spin when the spinner first lands on blue.

1. Find
   1. P(\(B = 4\))
   2. P(\(B \leq 5\))
   [4]
2. Find E(\(B^2\)) [3]

Steve invites Tamara to play a game with this spinner. Tamara must choose a colour, either red or blue. Steve will spin the spinner repeatedly until the spinner first lands on the colour Tamara has chosen. The random variable \(X\) represents the number of the spin when this occurs. If Tamara chooses red, her score is \(e^X\) If Tamara chooses blue, her score is \(X^2\)

1. State, giving your reasons and showing any calculations you have made, which colour you would recommend that Tamara chooses. [5]

The discrete random variable \(X\) has a geometric distribution. It is given that \(\text{Var}(X) = 20\). Determine \(P(X \geq 7)\). [6]