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.

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.
   1. State two assumptions needed for \(L\) to be well modelled by a geometric distribution.
   2. 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.

80\% of the residents of Kinwawa are in favour of a leisure centre being built in the town. 20 residents of Kinwawa are chosen at random and asked, in turn, whether they are in favour of the leisure centre.

1. Find the probability that more than 17 of these residents are in favour of the leisure centre. [3]
2. Find the probability that the 5th person asked is the first person who is not in favour of the leisure centre. [1]
3. Find the probability that the 7th person asked is the second person who is not in favour of the leisure centre. [2]