One unknown from sum constraint only

4 The number of fish, \(X\), caught by Pearl when she goes fishing can be modelled by the following discrete probability distribution:

1. Find the value of \(k\).
2. Find:
   1. \(\mathrm { E } ( X )\);
   2. \(\operatorname { Var } ( X )\).
3. When Pearl sells her fish, she earns a profit, in pounds, given by $$Y = 5 X + 2$$ Find:
   1. \(\mathrm { E } ( Y )\);
   2. the standard deviation of \(Y\).

1 The probability distribution for a discrete random variable \(X\) is given in the table below.

1. Find the value of \(c\).
2. Find the value of each of the following.
   The random variable \(Y\) is defined by \(Y = 2 X - 3\).
3. Find the value of each of the following.

1. The probability distribution for the prize money, \(\pounds X\) per ticket, in a local fundraising lottery is shown below.

1. Calculate the value of \(p\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
3. 1. What is the minimum lottery ticket price that the organiser should set in order to make a profit in the long run?
   2. Suggest why, in practice, people would be prepared to pay more than this minimum price.

A spinner is designed so that the score \(S\) is given by the following probability distribution.

1. Find the value of \(p\). [2]
2. Find \(\text{E}(S)\). [2]
3. Show that \(\text{E}(S^2) = 9.45\) [2]
4. Find \(\text{Var}(S)\). [2]

Tom and Jess play a game with this spinner. The spinner is spun repeatedly and \(S\) counters are awarded on the outcome of each spin. If \(S\) is even then Tom receives the counters and if \(S\) is odd then Jess receives them. The first player to collect 10 or more counters is the winner.

1. Find the probability that Jess wins after 2 spins. [2]
2. Find the probability that Tom wins after exactly 3 spins. [4]
3. Find the probability that Jess wins after exactly 3 spins. [3]

The table below shows the probability distribution for a discrete random variable \(X\).

\(x\)	0	1	2	3	4 or more
P(X = x)	0.35	0.25	\(k\)	0.14	0.1

Find the value of \(k\). Circle your answer. 0.14 \quad 0.16 \quad 0.18 \quad 1 [1 mark]

The probability distribution of a random variable \(X\) is given in the table.

1. Find the value of \(p\). [2]
2. Two values of \(X\) are chosen at random. Find the probability that the product of these values is 0. [3]

The probability distribution of the discrete random variable \(X\) is given in Fig. 4.

\(r\)	0	1	2	3	4
P\((X = r)\)	0.2	0.15	0.3	\(k\)	0.25

Fig. 4

1. Find the value of \(k\). [2]

\(X_1\) and \(X_2\) are two independent values of \(X\).

1. Find P\((X_1 + X_2 = 6)\). [3]