Four cards are drawn, without replacement, from a deck of 52 cards. Write the probability distribution for the number of red cards drawn

The probability distribution for the number of red cards drawn can be calculated as follows:

1. There are 26 red cards in a deck of 52 cards.

2. The possible outcomes for the number of red cards drawn are 0, 1, 2, 3, and 4.

3. The probability for each outcome can be calculated using the hypergeometric distribution formula:

   P(X=k) = [C(K, k) * C(N-K, n-k)] / C(N, n)

   where:

   • N is the total number of items (52 cards),
   • K is the total number of success states in the population (26 red cards),
   • n is the number of items drawn (4 cards), and
   • k is the number of success states drawn (the number of red cards drawn).

4. So, the probability distribution is:

   • P(X=0) = [C(26, 0) * C(52-26, 4-0)] / C(52, 4)
   • P(X=1) = [C(26, 1) * C(52-26, 4-1)] / C(52, 4)
   • P(X=2) = [C(26, 2) * C(52-26, 4-2)] / C(52, 4)
   • P(X=3) = [C(26, 3) * C(52-26, 4-3)] / C(52, 4)
   • P(X=4) = [C(26, 4) * C(52-26, 4-4)] / C(52, 4)

5. The values of these probabilities can be calculated using a calculator or statistical software.

The probability distribution for the number of red cards drawn can be calculated as follows:

1. There are 26 red cards in a deck of 52 cards.

2. The possible outcomes for the number of red cards drawn are 0, 1, 2, 3, and 4.

3. The probability of each outcome can be calculated using the hypergeometric distribution formula:

   P(X=k) = [C(K, k) * C(N-K, n-k)] / C(N, n)

   where:

   • N is the total number of items (52 cards),
   • K is the total number of success states in the population (26 red cards),
   • n is the number of items drawn (4 cards), and
   • k is the number of success states in the sample (the number of red cards drawn).

4. So, the probability distribution is:

   • P(X=0) = [C(26, 0) * C(26, 4)] / C(52, 4)
   • P(X=1) = [C(26, 1) * C(26, 3)] / C(52, 4)
   • P(X=2) = [C(26, 2) * C(26, 2)] / C(52, 4)
   • P(X=3) = [C(26, 3) * C(26, 1)] / C(52, 4)
   • P(X=4) = [C(26, 4) * C(26, 0)] / C(52, 4)

5. The values of these probabilities can be calculated using a calculator or a statistical software.

Note: C(a, b) denotes the number of combinations of a items taken b at a time.