Probability & Statistics 1 — Variant 2

An ordinary fair die is thrown repeatedly until a $5$ is obtained. The number of throws taken is denoted by the random variable $X$.

Write down the mean of $X$.

Find the probability that a $5$ is first obtained after the 3rd throw but before the 8th throw.

Find the probability that a $5$ is first obtained in fewer than $10$ throws.

The weights of bags of sugar are normally distributed with mean $1.04$ kg and standard deviation $\sigma$ kg. In a random sample of $2000$ bags of sugar, $72$ weighed more than $1.10$ kg. Find the value of $\sigma$.

On each day that Alexa goes to work, the probabilities that she travels by bus, by train or by car are $0.4$, $0.35$ and $0.25$ respectively. When she travels by bus, the probability that she arrives late is $0.55$. When she travels by train, the probability that she arrives late is $0.7$. When she travels by car, the probability that she arrives late is $x$. On a randomly chosen day when Alexa goes to work, the probability that she does not arrive late is $0.48$.

Find the value of $x$.

Find the probability that Alexa travels to work by train given that she arrives late.

A fair spinner has sides numbered $1$, $2$, $2$. Another fair spinner has sides numbered $-2$, $0$, $1$. Each spinner is spun. The number on the side on which a spinner comes to rest is noted. The random variable $X$ is the sum of the numbers for the two spinners.

Draw up the probability distribution table for $X$.

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

Every day Richard takes a flight between Astan and Bejin. On any day, the probability that the flight arrives early is $0.15$, the probability that it arrives on time is $0.55$ and the probability that it arrives late is $0.3$.

Find the probability that on each of $3$ randomly chosen days, Richard's flight does not arrive late.

Find the probability that for $9$ randomly chosen days, Richard's flight arrives early at least $3$ times.

$60$ days are chosen at random. Use an approximation to find the probability that Richard's flight arrives early at least $12$ times.