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

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

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)=\frac{\binom{26}{3}}{\binom{52}{3}}`

`P(B)=\frac{\binom{12}{3}}{\binom{52}{3}}`

`P(A\bigcap B)=\frac{\binom{6}{3}}{\binom{52}{3}}`

Hence,
`P(A\bigcup B)=\frac{\binom{26}{3}}{\binom{52}{3}}+\frac{\binom{12}{3}}{\binom{52}{3}}-\frac{\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

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

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* 4=12`.

Formula of probability is

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

`^nC_r=(n!)/(r!(n-r)!)`

Total outcomes is

`^(52)C_3=(52!)/(3!(52-3)!)=22100`

Total possible outcomes of getting 3 red is

`^(26)C_3=(26!)/(3!(26-3)!)=2600`

Total possible outcomes of getting 3 face is

`^(12)C_3=(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

`(2820)/(22100)=0.12760181\approx 0.1276`

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