Question

11.There are twelve face cards (the kings, queens, and jacks). What is the probability of picking four cards and getting at least one queen?

12.I remove the ace of spades from a deck of cards. In this modified deck, are kings and hearts independent events? Please explain.

Answer 1

solution:

11) Given data

Total no.of face cards = 12

out of 12 face cards

No.of queens = 4

No.of other cards = No.of kings + No.of jacks = 4

In the event of drawing 4 cards.we have

n(S) = 12C4 = 495

Let A = event that getting at least 1 queen

A' = event that getting 0 queen's

Here, n(A') = 8C4 = 70

Probability that getting at least 1 queen = P(A)

= 1 - Probability of getting 0 queen's

= 1 - P(A')

= 1 - n(A')/n(S)

= 1 - 70/495

= 425/495

= 0.8586

   Probability that getting at least 1 queen = 0.8586

12) Independent events : The probability of occurring of one event doesn't effect the probability of occurring another event.

For Independent events , P( AB) = P(A) * P(B)

when we remove the ace of spades from deck of cards,we have

Total no.of remaining cards = 52 - 1 = 51

when we draw one card from these cards

n(Sample space) = n(S) = 51C1 = 51

No.of kings = 4

No.of hearts = 13

Let A = event that drawn card is king

B = event that drawn card is a heart

AB = event that drawn card king of hearts

Here, n(A) = 4 ,  n(B) = 13 and n( AB) = 1

Probability of drawn card is king = P(A)

= n(A) / n(S)

= 4/51

Probability of drawn card is a heart = P(B)

= n(B) / n(S)

= 13/51

Probability of drawn card is a king of hearts = P( AB)

= n( AB) / n(S)

= 1/51

= 0.01961

Here observe that P(A) * P(B) = (4/51) * (13/51)

= 0.01999

!= P( AB)

Therefore, In the modified deck kings and hearts are not independent events