Final answer:
To create the probability distribution for flipping four coins, we set up a table with all possible outcomes of tails (0 to 4) and calculate their respective probabilities by dividing the number of ways each outcome can occur by the total of 16 possible outcomes.
Step-by-step explanation:
To create the probability distribution for the number of tails showing when flipping four coins, we need to consider all the possible outcomes (macrostates) and how many ways (microstates) each can occur. Since all microstates are equally likely in a fair coin toss, the probability of each macrostate is the number of microstates for that outcome divided by the total number of microstates (24 = 16).
The macrostates for a four-coin toss and their corresponding microstates are:
- 0 tails (4 heads): 1 way (HHHH)
- 1 tail: 4 ways (THHH, HTHH, HHTH, HHHT)
- 2 tails: 6 ways (TTHH, THTH, THHT, HTTH, HTHT, HHTT)
- 3 tails: 4 ways (TTTH, TTHT, THTT, HTTT)
- 4 tails (0 heads): 1 way (TTTT)
The probability distribution table (PDF) would look like this:
Number of Tails (X)Probability (P(X))01/1614/1626/1634/1641/16
To find probabilities, divide the number of ways by 16. For example, for 1 tail, P(X=1) = 4/16 = 0.25.