Probability & Statistics 1 — Variant 1

A summary of $20$ values of $x$ gives $\Sigma(x - 30) = 439$, $\Sigma(x - 30)^{2} = 12405$. A summary of another $25$ values of $x$ gives $\Sigma(x - 30) = 470$, $\Sigma(x - 30)^{2} = 11346$.

Find the mean of all $45$ values of $x$.

Find the standard deviation of all $45$ values of $x$.

In a certain area in the Arctic the probability that it snows on any given day is $0.7$, independent of all other days.

Find the probability that in a week ($7$ days) it snows on at least five days. A week in which it snows on at least five days out of seven is called a 'white' week.

Find the probability that in three randomly chosen weeks at least one is a white week. In a different area in the Arctic, the probability that a week is a white week is $0.8$.

Use a suitable approximation to find the probability that in $60$ randomly chosen weeks fewer than $47$ are white weeks.

Harry has three coins: One coin is biased so that the probability of obtaining a head when it is thrown is $\frac{1}{3}$. The second coin is biased so that the probability of obtaining a head when it is thrown is $\frac{1}{4}$. The third coin is biased so that the probability of obtaining a head when it is thrown is $\frac{1}{5}$. Harry throws the three coins. The random variable $X$ is the number of heads that he obtains.

Draw up the probability distribution table for $X$. Harry has two other coins, each of which is biased so that the probability of obtaining a head when it is thrown is $p$. He throws all five coins at the same time. The random variable $Y$ is the number of heads that he obtains.

Given that $P(Y = 0) = 6P(Y = 5)$, find the value of $p$.

The eight digits $1, 2, 2, 3, 4, 4, 4, 5$ are arranged in a line.

How many different arrangements are there of these $8$ digits?

Find the number of different arrangements of the $8$ digits in which there is a $2$ at the beginning, a $2$ at the end and the three $4$s are not all together. Three digits are selected at random from the eight digits $1, 2, 2, 3, 4, 4, 4, 5$.

Find the probability that the three digits are all different.