P(a ≤ X ≤ b) range probability

7 In a game,players attempt to score a goal by kicking a ball into a net.The probability that Leno scores a goal is 0.4 on any attempt,independently of all other attempts.The random variable \(X\) denotes the number of attempts that it takes Leno to score a goal.

1. Find \(\mathrm { P } ( X = 5 )\) .
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2. Find \(\mathrm { P } ( 3 \leqslant X \leqslant 7 )\) .
3. Find the probability that Leno scores his second goal on or before his 5th attempt. \includegraphics[max width=\textwidth, alt={}, center]{aeb7b26e-6754-4c61-b71e-e8169c617b91-10_2715_33_106_2017} \includegraphics[max width=\textwidth, alt={}, center]{aeb7b26e-6754-4c61-b71e-e8169c617b91-11_2723_33_99_22} Leno has 75 attempts to score a goal.
4. Use a suitable approximation to find the probability that Leno scores more than 28 goals but fewer than 35 goals.
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2 The random variable \(X\) has the distribution \(\operatorname { Geo } ( 0.2 )\). Find

1. \(\mathrm { P } ( X = 3 )\),
2. \(\mathrm { P } ( 3 \leqslant X \leqslant 5 )\),
3. \(\mathrm { P } ( X > 4 )\). Two independent values of \(X\) are found.
4. Find the probability that the total of these two values is 3 .

9 Each day Harry makes repeated attempts to light his gas fire. If the fire lights he makes no more attempts. On each attempt, the probability that the fire will light is 0.3 independent of all other attempts. Find the probability that

1. the fire lights on the 5th attempt,
2. Harry needs more than 1 attempt but fewer than 5 attempts to light the fire. If the fire does not light on the 6th attempt, Harry stops and the fire remains unlit.
3. Find the probability that, on a particular day, the fire lights.
4. Harry's week starts on Monday. Find the probability that, during a certain week, the first day on which the fire lights is Wednesday.

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]

A bag contains 3 green counters, 3 blue counters and \(w\) white counters. Counters are selected at random, one at a time, with replacement, until a white counter is drawn. The total number of counters selected, including the white counter, is denoted by \(X\).

1. In the case when \(w = 2\),
   1. write down the distribution of \(X\), [1]
   2. find \(P(3 < X \leq 7)\). [2]
2. In the case when E\((X) = 2\), determine the value of \(w\). [2]
3. In the case when \(w = 2\) and \(X = 6\), find the probability that the first five counters drawn alternate in colour. [2]