Find minimum n for P(X ≤ n) > threshold

9 Repeated independent trials of a certain experiment are carried out. On each trial the probability of success is 0.12 .

1. Find the smallest value of \(n\) such that the probability of at least one success in \(n\) trials is more than 0.95.
2. Find the probability that the 3rd success occurs on the 7th trial.

7 A fair die is thrown until a 6 appears for the first time. Assuming that the throws are independent, find

1. the probability that exactly 5 throws are needed,
2. the probability that fewer than 8 throws are needed,
3. the least integer \(n\) such that the probability of obtaining a 6 before the \(n\)th throw is at least 0.99 .

6 The score when two fair dice are thrown is the sum of the two numbers on the upper faces. Two fair dice are thrown repeatedly until a score of 6 is obtained. The number of throws taken is denoted by the random variable \(X\). Find the mean of \(X\). Find the least integer \(N\) such that the probability of obtaining a score of 6 in fewer than \(N\) throws is more than 0.95 .

6 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. Find the least integer \(N\) such that the probability of obtaining a 6 before the \(N\) th throw is more than 0.95.

7 A pair of fair coins is thrown repeatedly until a pair of tails is obtained. The number of throws taken is denoted by the random variable \(X\).

1. State the expected value of \(X\).
2. Find the probability that exactly 3 throws are required to obtain a pair of tails.
3. Find the probability that fewer than 4 throws are required to obtain a pair of tails.
4. Find the least integer \(N\) such that the probability of obtaining a pair of tails in fewer than \(N\) throws is more than 0.95 .

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

1. \(\mathrm { P } ( X = 5 )\),
2. \(\mathrm { P } ( X \geqslant 5 )\),
3. the least integer \(N\) such that \(\mathrm { P } ( X \leqslant N ) > 0.9995\).

6 In a skiing resort, for each day during the winter season, the probability that snow will fall on that day is 0.2 , independently of any other day. The first day of the winter season is 1 December. Find, for the winter season,

1. the probability that the first snow falls on 20 December,
2. the probability that the first snow falls before 5 December,
3. the earliest date in December such that the probability that the first snow falls on or before that date is at least 0.95 .

6 A fair die is thrown until a 5 or a 6 is obtained. The number of throws taken is denoted by the random variable \(X\). State the mean value of \(X\). Find the probability that obtaining a 5 or a 6 takes more than 8 throws. Find the least integer \(n\) such that the probability of obtaining a 5 or a 6 in fewer than \(n\) throws is more than 0.99.

3 In a large collection of coloured marbles of identical size, the proportion of green marbles is \(p\). One marble is chosen randomly, its colour is noted, and it is then replaced. This process is repeated until a green marble is chosen. The first green marble chosen is the \(X\) th marble chosen.

1. You are given that \(p = 0.3\).
   1. Find \(\mathrm { P } ( 5 \leqslant X \leqslant 10 )\).
   2. Determine the smallest value of \(n\) for which \(\mathrm { P } ( X = n ) < 0.1\).
2. You are given instead that \(\operatorname { Var } ( X ) = 42\). Determine the value of \(\mathrm { E } ( X )\).