Final answer:
To solve the problem, we listed out all possible outcomes of tossing a fair coin 4 times, counted the number of tails (T) for each outcome, and then calculated the probability of each number of tails showing up. We set up a probability distribution table for the random variable X, which represents the number of tails resulting from the tosses.
Step-by-step explanation:
A fair coin tossed 4 times gives us a total of 16 possible outcomes (24 = 16). To construct a probability distribution table for the random variable X, which denotes the number of tails shown up, we can list all possible outcomes and count the tails in each:
- 0 tails (TTTT): 1 way
- 1 tail (HTTT, THTT, TTHT, TTTH): 4 ways
- 2 tails (HTHT, HTTH, THHT, TTHH, HHTT, HTHH, THHH): 6 ways
- 3 tails (HHTH, HTHH, THHH): 4 ways
- 4 tails (HHHH): 1 way
Each outcome is equally likely, so the probability of each can be calculated by dividing the number of ways by the total number of outcomes (16). The probability distribution table for the number of tails (X) is as follows:
xP(X=x)01/1614/1626/1634/1641/16
We can also simplify these probabilities to 1/16, 1/4, 3/8, 1/4, and 1/16 respectively.