Final answer:
The probability distribution of the number of times the color orange is spun on a two-section spinner when spun three times has four possible values for the random variable X: 0, 1, 2, and 3, with corresponding probabilities 1/8, 3/8, 3/8, and 1/8 respectively.
Step-by-step explanation:
The question involves determining the probability distribution of the random variable X, which represents the number of times the color orange (O) appears when a spinner with equal sections of green and orange is spun three times. The sample space S consists of eight possible outcomes: {GGG, GGO, GOG, OGG, GOO, OGO, OOG, OOO}.
To find the probability distribution, we count the occurrences of O in each outcome:
- No O's: GGG (1 outcome)
- One O: GGO, GOG, OGG (3 outcomes)
- Two O's: GOO, OGO, OOG (3 outcomes)
- Three O's: OOO (1 outcome)
Since the spinner sections are equal, each outcome in the sample space has an equal chance of occurring, making the probability for each outcome 1/8. The probabilities for the number of times O appears are:
- P(X=0) = 1/8
- P(X=1) = 3/8
- P(X=2) = 3/8
- P(X=3) = 1/8
This is a probability distribution where the probability for each value of X is proportional to the number of outcomes in the sample space that correspond to that value. Out of the provided graphs (which we cannot see), the correct graph should have the probabilities of 0, 1/8; 1, 3/8; 2, 3/8; and 3, 1/8 plotted accordingly.