Question

g What is the probability of being dealt exactly three of a kind (like three kings or three 7’s, etc.) in a five card hand from a deck of 52 cards?

Answer 1

Answer: 0.02257

Step-by-step explanation:

Given : Total cards in a deck = 52

Number of ways to select any 5 cards : $^{52}C_5$

Since , there are total 13 kinds of card (includes Numbers from 2 to 9 and Ace , king, queen and jack).

Of each kind , there are 4 cards.

Number of ways to select three cards in a five card hand of a single kind : $^{4}C_3\times^{48}C_2$

Number of ways to select three cards in a five card hand of a exactly three of a kind : $13\times^{4}C_3\times^{48}C_2$

Now , the required probability = $\dfrac{13\times^{4}C_3\times^{48}C_2}{^{52}C_5}$

$=\dfrac{13\times4\times\dfrac{48!}{2!46!}}{\dfrac{52!}{5!47!}}\\\\\\=\dfrac{58656}{2598960}$

$=0.022569027611\approx0.02257$

∴ The probability of being dealt exactly three of a kind (like three kings or three 7’s, etc.) in a five card hand from a deck of 52 cards= 0.02257