Conditional probability with random variables

14 A random variable \(X\) has probability distribution given by \(\mathrm { P } ( X = x ) = \frac { 1 } { 860 } ( 1 + x )\) for \(x = 1,2,3 , \ldots , 40\).

1. Find \(\mathrm { P } ( X > 39 )\).
2. Given that \(x\) is even, determine \(\mathrm { P } ( X < 10 )\). \section*{END OF QUESTION PAPER}

1. Adana selects one number at random from the distribution of \(X\) which has the following probability distribution.

1. Given that the number selected by Adana is not 5 , write down the probability it is 0
2. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = 75\)
3. Find \(\operatorname { Var } ( X )\)
4. Find \(\operatorname { Var } ( 4 - 3 X )\) Bruno and Charlie each independently select one number at random from the distribution of \(X\)
5. Find the probability that the number Bruno selects is greater than the number Charlie selects. Devika multiplies Bruno's number by Charlie's number to obtain a product, \(D\)
6. Determine the probability distribution of \(D\)

3 The probability distribution of the discrete random variable \(X\) is defined as follows. $$\mathrm { P } ( X = x ) = k ( 2 + x ) ( 5 - x ) \quad \text { for } x = 0,1,2,3,4$$

1. Show that \(k = \frac { 1 } { 50 }\).
2. Find the variance of \(X\).
3. Find \(\mathrm { P } ( X = 4 \mid X > 0 )\).

Miguel has six numbered tiles, labelled 2, 2, 3, 3, 4, 4. He selects two tiles at random, without replacement. The variable \(M\) denotes the sum of the numbers on the two tiles.

1. Show that \(P(M = 6) = \frac{1}{3}\) [2]

The table shows the probability distribution of \(M\)

\(m\)	4	5	6	7	8
\(P(M = m)\)	\(\frac{1}{15}\)	\(\frac{4}{15}\)	\(\frac{1}{3}\)	\(\frac{4}{15}\)	\(\frac{1}{15}\)

Miguel returns the two tiles to the collection. Now Sofia selects two tiles at random from the six tiles, without replacement. The variable \(S\) denotes the sum of the numbers on the two tiles that Sofia selects.

1. Find \(P(M = S)\) [3]
2. Find \(P(S = 7 | M = S)\) [4]

Miguel has six numbered tiles, labelled 2, 2, 3, 3, 4, 4. He selects two tiles at random, without replacement. The variable \(M\) denotes the sum of the numbers on the two tiles.

1. Show that \(P(M = 6) = \frac{1}{3}\) [2]

The table shows the probability distribution of \(M\) Miguel returns the two tiles to the collection. Now Sofia selects two tiles at random from the six tiles, without replacement. The variable \(S\) denotes the sum of the numbers on the two tiles that Sofia selects.

1. Find \(P(M = S)\) [3]
2. Find \(P(S = 7 | M = S)\) [4]

A random variable \(X\) has probability distribution given by \(P(X = x) = \frac{1}{860}(1 + x)\) for \(x = 1, 2, 3, \ldots, 40\).

1. Find \(P(X > 39)\). [2]
2. Given that \(x\) is even, determine \(P(X < 10)\). [6]