Probability & Statistics 1 — Variant 2

$80\%$ of the residents of Kinwawa are in favour of a leisure centre being built in the town. 20 residents of Kinwawa are chosen at random and asked, in turn, whether they are in favour of the leisure centre.

Find the probability that more than $17$ of these residents are in favour of the leisure centre.

Find the probability that the 5th person asked is the first person who is not in favour of the leisure centre.

Find the probability that the 7th person asked is the second person who is not in favour of the leisure centre.

The probability that it will rain on any given day is $x$. If it is raining, the probability that Aran wears a hat is $0.8$ and if it is not raining, the probability that he wears a hat is $0.3$. Whether it is raining or not, if Aran wears a hat, the probability that he wears a scarf is $0.4$. If he does not wear a hat, the probability that he wears a scarf is $0.1$. The probability that on a randomly chosen day it is not raining and Aran is not wearing a hat or a scarf is $0.36$. Find the value of $x$.

Marco has four boxes labelled $K$, $L$, $M$ and $N$. He places them in a straight line in the order $K$, $L$, $M$, $N$ with $K$ on the left. Marco also has four coloured marbles: one is red, one is green, one is white and one is yellow. He places a single marble in each box, at random. Events $A$ and $B$ are defined as follows. $A$: The white marble is in either box $L$ or box $M$. $B$: The red marble is to the left of both the green marble and the yellow marble. Determine whether or not events $A$ and $B$ are independent.