A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability of two successes.

To find the probability of getting two successes (doublets) when a pair of dice is thrown four times, we can use the binomial probability formula.

Understanding the Basics

A doublet occurs when both dice show the same number. The possible doublets are (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6), giving us a total of 6 successful outcomes. Since there are 36 possible outcomes when rolling two dice, the probability of rolling a doublet in one throw is:

P(doublet) = 6/36 = 1/6

Setting Up the Problem

We are interested in the scenario where we roll the dice 4 times and want exactly 2 successes (doublets). The probability of not rolling a doublet is:

P(not doublet) = 1 - P(doublet) = 1 - 1/6 = 5/6

Applying the Binomial Formula

The binomial probability formula is:

P(X = k) = C(n, k) * (p^k) * (q^(n-k))

• C(n, k) is the number of combinations of n items taken k at a time.
• p is the probability of success (1/6).
• q is the probability of failure (5/6).
• n is the number of trials (4).
• k is the number of successes (2).

Calculating the Values

First, calculate the combinations:

C(4, 2) = 4! / (2!(4-2)!) = 6

Now plug the values into the formula:

P(X = 2) = 6 * (1/6)^2 * (5/6)^(4-2)

This simplifies to:

P(X = 2) = 6 * (1/36) * (25/36) = 6 * 25 / 1296 = 150 / 1296

Final Result

Thus, the probability of getting exactly two doublets when rolling a pair of dice four times is:

P(X = 2) = 25 / 216