Pick THREE cards from a dock of 52. Let the events A, B, C he as follows: A = \"
ID: 3172623 • Letter: P
Question
Pick THREE cards from a dock of 52. Let the events A, B, C he as follows: A = "at least two of the cards are aces" B = "at most two of the cards are aces" Remember that a deck of 52 playing cards contains 4 aces. There are 4 suits in a deck: clubs, hearts, diamonds and spades. Each suit has 13 cards. Find the number of possible outcomes in the events A and B. Find the probability of events A and B. In other words, find P[A] and P[B]. Are the events A and B mutually exclusive? Explain your reasoning. Are the events A and B collectively exhaustive? Explain your reasoning. Calculate the probability P[A union B].Explanation / Answer
a)
Event A = At least 2 of the cards are aces.
There are two possible chances here :
i) 2 out of 3 cards are aces.
ii) All the 3 cards are aces.
No. of Outcomes for 2 out of 3 cards to be ace = 4C2 * 48C1 = 6 * 48 = 288
No of Outcomes for all 3 cards to be ace = 4C3 * 48C0 = 4
Hence total outcomes for event A = 288 + 4 = 292
Event B = At most 2 of the cards are aces.
There are three possible chances here :
i) 1 out of 3 cards are aces.
ii) 2 out of 3 cards are aces.
iii) No ace out of 3 cards.
No. of Outcomes for 1 out of 3 cards to be ace = 4C1 * 48C2 = 4 * 1128 = 4512
No. of Outcomes for 2 out of 3 cards to be ace = 4C2 * 48C1 = 6 * 48 = 288
No of Outcomes for no card to be ace = 4C0 * 48C3 = 17296
Hence total outcomes for event A = 4512 + 288 + 17296 = 22096
b)
Total number of outcomes of picking 3 cards out of 52 = 52C3 = 22100
P[A] = Total number of outcomes in Event A / Total number of outcomes of picking 3 cards out of 52
= 292 / 22100
=0.0132
P [ B ] =Total number of outcomes in Event A / Total number of outcomes of picking 3 cards out of 52
= 22096 / 22100
= 0.9998
C)
Both events are not mutually exclusive as there are chances of both event A and B to occur simultaneously when exactly 2 out of 3 cards are aces.
d)
Both events are mutually exhaustive as there are chances of both event A and B to occur simultaneously when exactly 2 out of 3 cards are aces.