Step 1

Given,

A hand of 5 cards is dealt to each of three players from a standard deck of 52 cards.

Step 2

(a) The probability that one of the players receives all four Aces:

Let X be the number of Ace cards that a player receives

The probability of getting an Ace is \(\displaystyle{p}={\frac{{{4}}}{{{52}}}}={0.0769}\)

\(\displaystyle{n}={5}\)

\(\displaystyle{X}\sim{B}in{o}{m}{i}{a}{l}{\left({5},{0.0769}\right)}\)

\(\displaystyle{P}{\left({X}={4}\right)}={5}{C}_{{{4}}}{\left({0.0769}\right)}^{{{4}}}{\left({1}-{0.0769}\right)}^{{{5}-{4}}}\)

\(\displaystyle={0.00016}\)

(b) The probability that at least one player receives no hearts :

Let X be the number of heart cards that a player receives

The probability of getting a heart is \(\displaystyle{p}={\frac{{{13}}}{{{52}}}}={0.25}\)

\(\displaystyle{n}={5}\)

\(\displaystyle{X}\sim{B}in{o}{m}{i}{a}{l}{\left({5},{0.25}\right)}\)

The probability that no player receives no heart :

\(\displaystyle{P}{\left({X}={0}\right)}={5}{C}_{{{0}}}{\left({0.25}\right)}^{{{0}}}{\left({1}-{0.25}\right)}^{{{5}-{0}}}\)

\(\displaystyle={0.2373}\)

The probability that at least one player receives no hearts \(\displaystyle={1}-\text{No player receives no heart}\)

\(\displaystyle={1}-{0.2373}\)

\(\displaystyle={0.7627}\)