Question

QUESTION 2 Two cards are drawn at random from a standard deck of 52 cards, WITHOUT replacement. To three decimal places find the probability of the following: a) P(drawing a Jack) b) P(drawing a Seven given that a Jack has already been removed) c) P(drawing a Jack and then a Seven)

1. diemkieu
For a standard deck of 52 cards, the required probabilities are:
a) P(drawing a Jack) = 0.005
b) P(drawing a Seven given that a Jack has already been removed) =  0.078
c) P(drawing a Jack and then a Seven) = 0.006
For given question,
We have been given two cards are drawn at random from a standard deck of 52 cards, without replacement.
We need to find the required probabilities.
We know that the probabilities are used to determine the chances of an event.
Formula of the probability of an event A is:
P(A) = n(A)/n(S)
where,  n(A) is the number of favorable outcomes, n(S) is the total number of events in the sample space.
Here, n(S) = 52
(a) When two cards are drawn at random, we need to find the P(drawing a Jack)
In a standard deck of 52 cards, there are  4 jacks.
⇒ P(J) = 4/52
After removing a one Jack card there would be 51 cards in a deck.
So, the probability of drawing two Jack cards without replacement would be,
⇒ P(JJ) = 4/52 × 3/51
⇒ P(JJ) = 1/221
⇒ P(JJ) = 0.005
So, P(drawing a Jack) = 0.005
(b)  When two cards are drawn at random, we need to find the probability of drawing a Seven given that a Jack has already been removed.
There are 4 Seven in a standard deck of 52 cards.
Here, Jack has already been removed
So, n(S) = 51
⇒ P(drawing a Seven given that a Jack has already been removed) = 4/51
⇒ P(drawing a Seven given that a Jack has already been removed) = 0.078
(c)  When two cards are drawn at random, we need to find the probability that drawing a Jack and then a Seven.
P(drawing a Jack and then a Seven)
= P(drawing a Jack) × P(drawing a Seven given that a Jack has already been removed)
So, P(drawing a Jack and then a Seven) = 4/52 × 4/51
⇒ P(drawing a Jack and then a Seven) = 1/13 × 4/51
⇒ P(drawing a Jack and then a Seven) = 4/663
⇒ P(drawing a Jack and then a Seven) = 0.006
Therefore, for a standard deck of 52 cards, the required probabilities are:
a) P(drawing a Jack) = 0.005
b) P(drawing a Seven given that a Jack has already been removed) =  0.078
c) P(drawing a Jack and then a Seven) = 0.006