Sum or product of two independent values

6 The probability distribution for a random variable \(Y\) is shown in the table.

1. Calculate \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\). Another random variable, \(Z\), is independent of \(Y\). The probability distribution for \(Z\) is shown in the table.

   \(z\)	1	2	3
   \(\mathrm { P } ( Z = z )\)	0.1	0.25	0.65

   One value of \(Y\) and one value of \(Z\) are chosen at random. Find the probability that
2. \(Y + Z = 3\),
3. \(Y \times Z\) is even.

5. Manon has two biased spinners, one red and one green. The random variable \(R\) represents the score when the red spinner is spun.
The random variable \(G\) represents the score when the green spinner is spun.
The probability distributions for \(R\) and \(G\) are given below. Manon spins each spinner once and adds the two scores.

1. Find the probability that
   1. the sum of the two scores is 7
   2. the sum of the two scores is less than 4 The random variable \(X = m R + n G\) where \(m\) and \(n\) are integers. $$\mathrm { P } ( X = 20 ) = \frac { 1 } { 6 } \quad \text { and } \quad \mathrm { P } ( X = 50 ) = \frac { 1 } { 4 }$$
2. Find the value of \(m\) and the value of \(n\)

1. Two spinners are in the form of an equilateral triangle, whose three regions are labelled 1,2 and 3, and a square, whose four regions are labelled \(1,2,3\) and 4 . Both spinners are biased and the probability distributions for the scores \(X\) and \(Y\) obtained when they are spun are respectively:

1. Find the values of \(p\) and \(q\).
2. Find the probability that, when the two spinners are spun together, the sum of the two scores is (i) 5, (ii) less than 4 .
3. State an assumption that you have made in answering part (b) and explain why it is likely to be justifiable.
4. Calculate \(\mathrm { E } ( X + Y )\).

12 Rob has two six-sided dice, each with sides numbered 1, 2, 3, 4, 5, 6.
One dice is fair. The other dice is biased, with probabilities as shown in the table. Rob throws each dice once and notes the two scores, \(X\) on the fair dice and \(Y\) on the biased dice. He then calculates the value of the variable \(S\) which is defined as follows.

• If \(X \leqslant 3\), then \(S = X + 2 Y\).
• If \(X > 3\), then \(S = X + Y\).
  1. (a) Draw up a sample space diagram showing all the possible outcomes and the corresponding values of \(S\).
     (b) On your diagram, circle the four cells where the value \(S = 10\) occurs.
  2. Explain the mistake in the following calculation.

$$\mathrm { P } ( S = 10 ) = \frac { \text { Number of outcomes giving } S = 10 } { \text { Total number of outcomes } } = \frac { 4 } { 36 } = \frac { 1 } { 9 } .$$

Find the correct value of \(\mathrm { P } ( S = 10 )\).

Given that \(S = 10\), find the probability that the score on one of the dice is 4 .

The events " \(X = 1\) or 2 " and " \(S = n\) " are mutually exclusive. Given that \(\mathrm { P } ( S = n ) \neq 0\), find the value of \(n\).

The number of flowers which grow on a certain type of plant can be modelled by the discrete random variable \(X\) The probability distribution of \(X\) is given in the table below.

\(x\)	0	1	2	3	4	5 or more
P(\(X = x\))	0.03	0.15	0.22	0.31	0.09	\(p\)

1. Find the value of \(p\) [2 marks]
2. Two plants of this type are randomly selected from a large batch received from a local garden centre. Find the probability that the two plants will produce a total of three flowers. [3 marks]
3. 1. State one assumption necessary for the calculation in part (b) to be valid. [1 mark]
   2. Comment on whether, in reality, this assumption is likely to be valid. [1 mark]

The discrete random variable \(D\) has the following probability distribution where \(k\) is a constant.

1. Show that the value of \(k\) is \(\frac{600}{137}\) [2]
2. The random variables \(D_1\) and \(D_2\) are independent and each have the same distribution as \(D\). Find \(P(D_1 + D_2 = 80)\) Give your answer to 3 significant figures. [3]
3. A single observation of \(D\) is made. The value obtained, \(d\), is the common difference of an arithmetic sequence. The first 4 terms of this arithmetic sequence are the angles, measured in degrees, of quadrilateral \(Q\) Find the exact probability that the smallest angle of \(Q\) is more than \(50°\) [5]

The discrete random variable \(X\) takes values 1, 2, 3, 4 and 5, and its probability distribution is defined as follows. $$\mathrm{P}(X = x) = \begin{cases} a & x = 1, \\ \frac{1}{2}\mathrm{P}(X = x - 1) & x = 2, 3, 4, 5, \\ 0 & \text{otherwise,} \end{cases}$$ where \(a\) is a constant.

1. Show that \(a = \frac{16}{31}\). [2]

The discrete probability distribution for \(X\) is given in the table.

1. Find the probability that \(X\) is odd. [1]

Two independent values of \(X\) are chosen, and their sum \(S\) is found.

1. Find the probability that \(S\) is odd. [2]
2. Find the probability that \(S\) is greater than 8, given that \(S\) is odd. [3]

Sheila sometimes needs several attempts to start her car in the morning. She models the number of attempts she needs by the discrete random variable \(Y\) defined as follows. $$\mathrm{P}(Y = y + 1) = \frac{1}{2}\mathrm{P}(Y = y) \quad \text{for all positive integers } y.$$

1. Find P\((Y = 1)\). [2]
2. Give a reason why one of the variables, \(X\) or \(Y\), might be more appropriate as a model for the number of attempts that Sheila needs to start her car. [1]