If five coins are flipped, and we define Y as the number of tails that appear, then list all the possible outcomes or sample space. Construct the probability distribution, and draw the histogram.
Answer (1)
Answer:Here's how to solve this problem step-by-step: 1. Sample Space: The sample space represents all possible outcomes when flipping five coins. Each coin can be either heads (H) or tails (T). We can represent the outcomes as sequences of H and T, like this: - HHHHH- HHHHT- HHHTH- HHHTT- HHTHH- HHTHT- HHTTH- HHTTT- HTHHH- HTHHT- HTHTH- HTHTT- HTTHH- HTTHT- HTTTH- HTTTT- THHHH- THHHT- THHTH- THHTT- THTHH- THTHT- THTTH- THTTT- TTHHH- TTHHT- TTHTH- TTHTT- TTTHH- TTTHT- TTTTH- TTTTT There are a total of 2⁵ = 32 possible outcomes. 2. Probability Distribution: The probability distribution shows the probability of getting each possible number of tails (Y). We can calculate this using the binomial probability formula: P(Y=k) = (nCk) * p^k * (1-p)^(n-k) Where: - n = number of trials (coin flips) = 5- k = number of successes (tails) = 0, 1, 2, 3, 4, or 5- p = probability of success (getting tails) = 0.5- nCk = the number of combinations of n items taken k at a time (also written as ⁵Cₖ or ₅Cₖ) Let's calculate the probabilities: Y (Number of Tails) P(Y) Calculation 0 1/32 (5C0) * (0.5)^0 * (0.5)^5 1 5/32 (5C1) * (0.5)^1 * (0.5)^4 2 10/32 (5C2) * (0.5)^2 * (0.5)^3 3 10/32 (5C3) * (0.5)^3 * (0.5)^2 4 5/32 (5C4) * (0.5)^4 * (0.5)^1 5 1/32 (5C5) * (0.5)^5 * (0.5)^0 3. Histogram: A histogram visually represents the probability distribution. The horizontal axis shows the number of tails (Y), and the vertical axis shows the probability P(Y). Each bar represents the probability of a specific number of tails. Because creating a visual histogram here is difficult, I recommend using a spreadsheet program (like Excel or Google Sheets) or a graphing calculator to create the histogram based on the probability distribution table above. The bars should have heights corresponding to the probabilities calculated. The histogram will be symmetrical, with the highest bar at Y=2 and Y=3.
