Question

3 cards are drawn from a standard deck without replacement. What is the probability that at least one of the cards drawn is a red card?

Answer 1

Answer: $\dfrac{15}{17}$

Step-by-step explanation:

Total number of cards in a deck = 52

Number of red cards = 26

Number of cards not red =

Number of ways to draw not red cards = $^{26}C_3$

Total ways to draw 3 cards = $^{52}C_3$

The probability that none of three cards are red = $\dfrac{^{26}C_3}{^{52}C_3}$

$=\dfrac{\dfrac{26!}{3!(26-3)!}}{\dfrac{52!}{3!(52-3)!}}$  [∵ $^nC_r=\dfrac{n!}{r!(n-r)!}$]

$=\dfrac{\dfrac{26\times25\times24\times23!}{(23)!}}{\dfrac{52\times51\times50\times49!}{3!(49)!}}=\dfrac{2}{17}$

Now , the probability that at least one of the cards drawn is a red card = 1- Probability that none cards are red

$=1-\dfrac{2}{17}=\dfrac{17-2}{17}=\dfrac{15}{17}$

Hence, the required probability = $\dfrac{15}{17}$