Probability & Statistics 1 — Variant 3

Becky sometimes works in an office and sometimes works at home. The random variable $X$ denotes the number of days that she works at home in any given week. It is given that $\mathrm{P}(X = x) = kx(x + 1)$, where $k$ is a constant and $x = 1, 2, 3 \text{ or } 4$ only.

Draw up the probability distribution table for $X$, giving the probabilities as numerical fractions.

Find $\mathrm{E}(X)$ and $\mathrm{Var}(X)$.

Tim has two bags of marbles, $A$ and $B$. Bag $A$ contains 8 white, 4 red and 3 yellow marbles. Bag $B$ contains 6 white, 7 red and 2 yellow marbles. Tim also has an ordinary fair 6-sided dice. He rolls the dice. If he obtains a 1 or 2, he chooses two marbles at random from bag $A$, without replacement. If he obtains a 3, 4, 5 or 6, he chooses two marbles at random from bag $B$, without replacement.

Find the probability that both marbles are white.

Find the probability that the two marbles come from bag $B$ given that one is white and one is red.

Jai and his wife Kaz are having a party. Jai has invited five friends and each friend will bring his wife.

At the beginning of the party, the 12 people will stand in a line for a photograph.

How many different arrangements are there of the 12 people if Jai stands next to Kaz and each friend stands next to his own wife?

How many different arrangements are there of the 12 people if Jai and Kaz occupy the two middle positions in the line, with Jai's five friends on one side and the five wives of the friends on the other side?

For a competition during the party, the 12 people are divided at random into a group of 5, a group of 4 and a group of 3. Find the probability that Jai and Kaz are in the same group as each other.