Geometric distribution (first success)

2 An ordinary fair die is thrown repeatedly until a 1 or a 6 is obtained.

1. Find the probability that it takes at least 3 throws but no more than 5 throws to obtain a 1 or a 6 .
   On another occasion, this die is thrown 3 times. The random variable \(X\) is the number of times that a 1 or a 6 is obtained.
2. Draw up the probability distribution table for \(X\).
3. Find \(\mathrm { E } ( X )\).

7 Passengers are travelling to Picton by minibus. The probability that each passenger carries a backpack is 0.65 , independently of other passengers. Each minibus has seats for 12 passengers.

1. Find the probability that, in a full minibus travelling to Picton, between 8 passengers and 10 passengers inclusive carry a backpack.
2. Passengers get on to an empty minibus. Find the probability that the fourth passenger who gets on to the minibus will be the first to be carrying a backpack.
3. Find the probability that, of a random sample of 250 full minibuses travelling to Picton, more than 54 will contain exactly 7 passengers carrying backpacks.

9 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. (a) Find the probability that \(X = 10\).
   (b) Find the probability that \(X < 10\).
2. Find the expected number of steps taken by the token.
3. Hence, write down the value of \(\mathrm { E } ( X )\).

1. A fair coin is tossed 4 times.

Find the probability that

1. an equal number of head and tails occur
2. all the outcomes are the same,
3. the first tail occurs on the third throw.

1 Bill and Gill send letters to potential sponsors of a show. On past experience, they know that \(5 \%\) of letters receive a favourable reply.

1. Bill sends a letter to each of 40 potential sponsors. Assuming that the number \(N\) of favourable responses can be modelled by a binomial distribution, find the mean and variance of \(N\).
2. Gill sends one letter at a time to potential sponsors. \(L\) is the number of letters she sends, up to and including the first letter that receives a favourable response.
   (a) State two assumptions needed for \(L\) to be well modelled by a geometric distribution.
   (b) Using the assumptions in part (ii)(a), find the smallest number of letters that Gill has to send in order to have at least a \(90 \%\) chance of receiving at least one favourable reply.

5 In a certain region, the probability that any given day in October is wet is 0.16 , independently of other days.

1. Find the probability that, in a 10-day period in October, fewer than 3 days will be wet.
2. Find the probability that the first wet day in October is 8 October.
3. For 4 randomly chosen years, find the probability that in exactly 1 of these years the first wet day in October is 8 October.