Question

Three cards are chosen at random from a standard deck of 52.

1. What is the probability that all three cards are hearts?

2. What is the probability that all three cards are even?

3. What is the probability that all three cards are hearts and even?

4. What is the probability that all three cards are not aces?

e) What is the probability of at least one ace?

Answer 1

solution:

Totan No.of Cards = 52

In the event of Three cards are drawn - n(S) = 52C3 = 22100

a) Let H = event of drawn 3 cards are hearts

No.of ways possible to take 3 from 13 hearts = n(H) =13C3 = 286

P(All 3 cards are hearts) = P(H) = n(H)/ n(S) = 286 / 22100 =0.0129

b) Let E = event of drawn cards are even

Total no.of even cards from deck of cards = 20

   No.of ways possible to take 3 from 20 even cards = n(E) =20C3 = 1140

   P(All 3 cards are even) = P(E) = n(E)/ n(S) = 1140 / 22100 =0.0516

c) Let E H = event of 3 cards are even and hearts

No.of even and heart cards = 5

n( E H ) = 5C3  =10

    P(All 3 cards are even and hearts) = n( E H ) / n(S) = 10 / 22100 = 0.00045

d) Let A = event of 3 cards are aces

Total No.of aces from deck of cards = 4

n(A) = 4C3  =4

    P(All 3 cards are aces) = n(A) / n(S) = 4 / 22100 = 0.0001809

P(All 3 cards are not aces) = 1 - P(All 3 cards are aces) = 1 - 0.00018 =0.9998

e) Let B = event of 3 cards at least one ace

= greater than or equal to one ace

= 1 ace + 2 ace + 3 ace

No.of aces =4 ,remaining =48

n(B) = 4C1 * 48C2 + 4C2 * 48C1 + 4C3  = 4 * 1035 + 6 *48 + 4 = 4432

   P(In 3 cards atleast 1 ace) = n(B) / n(S) = 4432 / 22100 = 0.2005