Question

Two card are drawn at random from a well-shuffled deck of 52 cards what is the probability that either both are red or both are queens

Answer 1

Answer:

The probability that both are red or queen is $\dfrac{331}{1326}$

Step-by-step explanation:

Given as :

Two card are drawn at random from a well-shuffled deck of 52 cards

Let The probability that both are red or queen = p(E)

2 card out of 52 cards = n(S) = $_{2}^{52}\textrm{C}$

Or, n(S) = $\dfrac{\angle 52}{\angle 2\times \angle 50}$

Or, n(S) = $\dfrac{52\times 51}{2}$ = 26 ×  51 = 1326

number of selection of card which is either red or queen = n(E)

i.e n(E) = $_{2}^{26}\textrm{C}$  +  $_{2}^{4}\textrm{C}$

or, n(E) = $\dfrac{\angle 26}{\angle 2\times \angle 24}$  + $\dfrac{\angle 4}{\angle 2\times \angle 2}$

Or , n(E) = $\dfrac{26\times 25}{2}$  +  $\dfrac{4\times 3}{2}$

∴     n(E) = 325 + 6 = 331

Now, required probability = p(E) = $\dfrac{n(E)}{n(S)}$

Or, p(E) = $\dfrac{331}{1326}$

∴   p(E) = $\dfrac{331}{1326}$

Hence, The probability that both are red or queen is $\dfrac{331}{1326}$  Answer

Answer 2

Probability that either both are red or both are queens is 0.2488.

Step-by-step explanation:

We are given that two card are drawn at random from a well-shuffled deck of 52 cards.

Total number of red cards = 26

Total number of queens = 4

Number of cards which are red as well as queen = 2

Total cards in deck = 52

Let event that card is red card = A

event that card is queen = B

So, probability that either both are red or both are queens =  $P(A \bigcup B)$

              $P(A \bigcup B) = P(A) + P(B) - P(A \bigcap B)$

P(A)  =  $\frac{^{26}C_2 }{^{52}C_2}$  =  $\frac{25}{102}$

P(B)  =  $\frac{^{4}C_2 }{^{52}C_2}$  =  $\frac{1}{221}$

$P(A \bigcap B)$  =  $\frac{^{2}C_2 }{^{52}C_2}$  =  $\frac{1}{1326}$

SO,  $P(A \bigcup B) = \frac{25}{102}+\frac{1}{221}-\frac{1}{1326}$

                        =  $\frac{55}{221}$  =  0.2488

Hence, the probability that either both are red or both are queens is 0.2488.