Question

We draw one card at a time with replacement from the top of a shuffled standard deck of cards and stop when we draw a Jack of any suit. Let X be the number of cards we draw. What is P( X $\leq$ 4 ) ? Round to two decimals.

Reminder: This is a sequence of Bernoulli trials but it is not a Binomial distribution. The distribution is the number of trails before the first "success"

Answer 1

The probability of drawing a Jack within four draws from a shuffled deck with replacement is approximately 0.63.

Step-by-step explanation:

To calculate P(X ≤ 4), which is the probability of drawing a Jack within four draws from a standard deck of cards with replacement, we recognize that each draw is a Bernoulli trial because it results in either a 'success' (drawing a Jack) or a 'failure' (not drawing a Jack). Since the deck is replaced after each draw, the trials are independent and the probability of success remains constant. In a standard deck, there are 4 Jacks out of 52 cards, so the probability of drawing a Jack (success) on a single trial is 4/52 or 1/13, and the probability of not drawing a Jack (failure) is 12/13.

To find P(X ≤ 4), we need to consider the probabilities of all scenarios where a Jack is drawn on or before the fourth card:

• P(X = 1) is the probability of drawing a Jack on the first draw only.
• P(X = 2) includes the probability of not drawing a Jack on the first draw but drawing one on the second draw.
• P(X = 3) accounts for not drawing a Jack on the first two draws but drawing one on the third draw.
• P(X = 4) considers not drawing a Jack on the first three draws but drawing one on the fourth.

Since these are mutually exclusive events, we add their probabilities to find the cumulative probability:

P(X ≤ 4) = P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = 1/13 + (12/13)*(1/13) + (12/13)^2*(1/13) + (12/13)^3*(1/13)

After calculating the sum of these probabilities, we get P(X ≤ 4) ≈ 0.63 (rounded to two decimals).