Problem $1.($Birthday paradox$)$ a$)$Find the probability that the birthdays of $12$ people fall on different months of the year. b$)$ What is the probability that, in a set of n randomly chosen people, at least two will share a birthday. Find the smallest n for which this probability is more than $50\%($a year is considered to consist of $365$ days$).$

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Problem $2.($Employee of the month$)$ The cloakroom attendant gave number plates to four people who had deposited their hats in the cloakroom at the same time. Then they confused all the hats and hung them up at random. Find the probability of the following events: a$)$ Each of the four persons will get his own hat; b$)$ Exactly three persons will get their hats; c$)$ Exactly two persons will get their hats; d$)$ Only one person will get his hat; e$)$ Nobody will get his hat.

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Problem $3.($Sharpshooters$)$ Two marksmen independently from each other shoot at the same target, each firing one shot. The probability of hitting the target for the first shooter is $0\mathrm{.}8,$ for the second $-0\mathrm{.}4.$ After that, one hole is detected at the target. Find the probability that the first shooter hit the target.

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