Log inSign up
Statistics · Question of the Day

Statistics Question of the Day

A fresh daily question to build accuracy, reinforce recall, and turn practice into a steady habit.
Friday, October 23, 2026

A fair coin is flipped 3 times. The sample space consists of the 8 equally likely sequences (e.g., HHT, TTT). Define the random variable XX as the number of heads minus the number of tails in the 3 flips. (For example, HHT gives X=1X=1 because 2−1=12-1=1.) Which table correctly represents the theoretical probability distribution of XX?

Keep practicing Statistics

Question of the Day

Answer today's Statistics question, reveal the full explanation, then keep the streak going with a new question every day.

A fair coin is flipped 3 times. The sample space consists of the 8 equally likely sequences (e.g., HHT, TTT). Define the random variable XX as the number of heads minus the number of tails in the 3 flips. (For example, HHT gives X=1X=1 because 2−1=12-1=1.) Which table correctly represents the theoretical probability distribution of XX?

  1. P(X=−3)=18, P(X=−1)=38, P(X=1)=38, P(X=3)=18P(X=-3)=\dfrac{1}{8},\ P(X=-1)=\dfrac{3}{8},\ P(X=1)=\dfrac{3}{8},\ P(X=3)=\dfrac{1}{8} (correct answer)
  2. P(X=−3)=14, P(X=−1)=14, P(X=1)=14, P(X=3)=14P(X=-3)=\dfrac{1}{4},\ P(X=-1)=\dfrac{1}{4},\ P(X=1)=\dfrac{1}{4},\ P(X=3)=\dfrac{1}{4}
  3. P(X=−3)=18, P(X=−1)=38, P(X=1)=18, P(X=3)=38P(X=-3)=\dfrac{1}{8},\ P(X=-1)=\dfrac{3}{8},\ P(X=1)=\dfrac{1}{8},\ P(X=3)=\dfrac{3}{8}
  4. P(HHH)=18, P(HHT)=18, P(HTH)=18,…P(\text{HHH})=\dfrac{1}{8},\ P(\text{HHT})=\dfrac{1}{8},\ P(\text{HTH})=\dfrac{1}{8},\ldots

Explanation: This problem involves finding the theoretical probability distribution of X = (number of heads) - (number of tails) when flipping a fair coin 3 times. The sample space contains 8 equally likely sequences, each with probability 1/8. To find X for each outcome, we count heads minus tails: HHH gives X = 3-0 = 3, HHT gives X = 2-1 = 1, HTH gives X = 2-1 = 1, HTT gives X = 1-2 = -1, THH gives X = 2-1 = 1, THT gives X = 1-2 = -1, TTH gives X = 1-2 = -1, and TTT gives X = 0-3 = -3. To find P(X = x), we count how many outcomes map to each x-value: P(X = -3) = 1/8 (only TTT), P(X = -1) = 3/8 (HTT, THT, TTH), P(X = 1) = 3/8 (HHT, HTH, THH), and P(X = 3) = 1/8 (only HHH). A common misconception is thinking the X-values themselves are equally likely, when in fact we must count how many coin sequences produce each X-value.