Question

3 cards are drawn from a standard deck of 52 cards. Find the probability of drawing either 3 red cards or 3 face cards.

Answer 1

Answer:

The probability of drawing either 3 red cards or 3 face cards is: 0.1267

Step-by-step explanation:

We are given a deck of 52 playing cards out of which we draw 3 cards.

We are asked to find the probability that the 3 card drawn are either 3 red cards or 3 face cards.

We know that in a playing cards we have 26 red cards and 12 face cards.

so we have to choose our card either from 26 red cards or 12 face cards and there are 6 cards which are both red card and is a face as well.

Let P denote the probability of an event.

Let A be the set that the card drawn is red.

B denote the set that the card drawn is a face card.

A∩B denote the set that the card drawn is red as well as is a face card.

We need to find the probability that the card drawn is either 3 red cards or 3 face cards i.e. we are asked to find P(A∪B)

We know that

P(A∪B)=P(A)+P(B)-P(A∩B)

Now  $P(A)=\dfrac{\binom{26}{3}}{\binom{52}{3}}$

$P(B)=\dfrac{\binom{12}{3}}{\binom{52}{3}}$

$P(A\bigcap B)=\dfrac{\binom{6}{3}}{\binom{52}{3}}$

Hence, $P(A\bigcup B)=\dfrac{\binom{26}{3}}{\binom{52}{3}}+\dfrac{\binom{12}{3}}{\binom{52}{3}}-\dfrac{\binom{6}{3}}{\binom{52}{3}}$

$P(A\bigcup B)=0.1267$

Hence, the probability of drawing either 3 red cards or 3 face cards is: 0.1267

Answer 2

Answer:

The probability of drawing either 3 red cards or 3 face cards is 0.1276.

Step-by-step explanation:

Total number of card in standard deck is 52.

Number of red cards is 26.

Face card are king, queen and jack. Total number of face cards are $3\times 4=12$.

Formula of probability is

$P=\frac{\text{Number of possible outcomes}}{\text{Total outcomes}}$

$^nC_r=\frac{n!}{r!(n-r)!}$

Total outcomes is

$^{52}C_3=\frac{52!}{3!(52-3)!}=22100$

Total possible outcomes of getting 3 red is

$^{26}C_3=\frac{26!}{3!(26-3)!}=2600$

Total possible outcomes of getting 3 face is

$^{12}C_3=\frac{12!}{3!(12-3)!}=220$

Total possible outcomes of getting either 3 black or 3 face cards is

$2600+220=2820$

probability of drawing either 3 red cards or 3 face cards is

$\frac{2820}{22100}=0.12760181\approx 0.1276$

probability of drawing either 3 red cards or 3 face cards is 0.1276.