Determine p from given mean or variance

9 At a ski resort, the probability of snow on any particular day is constant and equal to \(p\). The skiing season begins on 1 November. The random variable \(X\) denotes the day of the skiing season on which the first snowfall occurs. (For example, if the first snowfall is on 5 November, then \(X = 5\).) The variance of \(X\) is \(\frac { 4 } { 9 }\).

1. Show that \(4 p ^ { 2 } + 9 p - 9 = 0\) and hence find the value of \(p\).
2. Find the probability that the first snowfall will be on 3 November.
3. Find the probability that the first snowfall will not be before 4 November.
4. Find the least integer \(N\) so that the probability of the first snowfall being on or before the \(N\) th day of November is more than 0.999 .

7 The probability that a driver passes an advanced driving test has a fixed value \(p\) for each attempt. A driver keeps taking the test until he passes. The random variable \(X\) denotes the number of attempts required for the driver to pass. The variance of \(X\) is 3.75 .

1. Show that \(15 p ^ { 2 } + 4 p - 4 = 0\) and hence find the value of \(p\).
2. Find \(\mathrm { P } ( X = 5 )\).
3. Find \(\mathrm { P } ( 3 \leqslant X \leqslant 7 )\).

6 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 \(\frac { 2 } { 3 }\). Find

1. \(\mathrm { P } ( X = 4 )\),
2. \(\mathrm { P } ( X > 4 )\),
3. the least integer \(N\) such that \(\mathrm { P } ( X \leqslant N ) > 0.999\).

8 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. Find the probability that Lan first gets a seat on Monday of the second week in his new job.
3. Find the least integer \(N\) such that \(\mathrm { P } ( X \leqslant N ) > 0.9\), and identify the day and the week that corresponds to this value of \(N\).

6 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 \(X\).

1. Show that the probability that a head is obtained when the coin is tossed once is \(\frac { 2 } { 3 }\).
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2. Find \(\mathrm { P } ( X = 4 )\).
3. Find \(\mathrm { P } ( X > 4 )\).
4. Find the least integer \(N\) such that \(\mathrm { P } ( X \leqslant N ) > 0.999\).

7 A town council is planning to introduce a new set of parking regulations. An interviewer contacts randomly chosen people in the town and asks them whether they are in favour of the proposal. The first person who is not in favour of the regulation is the \(R\) th person interviewed. It can be assumed that the probability that any randomly chosen person is not in favour of the proposal is a constant \(p\), and that \(p\) does not equal 0 or 1 . Assume first that \(\mathrm { E } ( R ) = 10\).

1. Determine \(\mathrm { P } ( R \geqslant 14 )\). Now, without the assumption that \(\mathrm { E } ( R ) = 10\), consider a general value of \(p\).
   It is given that \(\mathrm { P } ( R = 3 ) - 0.4 \times \mathrm { P } ( R = 2 ) - a \times \mathrm { P } ( R = 1 ) = 0\), where \(a\) is a positive constant.
2. Determine the range of possible values of \(a\).

\(\mathbf { 4 }\) & \(\mathbf { ( a ) }\) & Geometric & M1 & 1.1 & Stated explicitly
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