Which of the following experiments have equally likely outcomes? Explain. (i) A driver attempts to start a car. The car starts or does not start. (ii) A player attempts to shoot a basketball. She/he shoots or misses the shot. (iii) A trial is made to answer a true-false question. The answer is right or wrong. (iv) A baby is born. It is a boy or a girl.
Step-by-Step Solution
Key Concept: An experiment has equally likely outcomes when each possible outcome has the same probability of occurrence. For a binary experiment this means the probability of each of the two outcomes must be \(\frac{1}{2}\).
1. Identify the outcomes for each experiment and denote their probabilities.
- (i) Outcomes: "Car starts" (S) and "Car does not start" (N). Let \(P(S)=p\) and \(P(N)=1-p\).
- (ii) Outcomes: "Shot made" (M) and "Shot missed" (X). Let \(P(M)=q\) and \(P(X)=1-q\).
- (iii) Outcomes: "Answer right" (R) and "Answer wrong" (W). If the student guesses randomly, \(P(R)=\frac{1}{2}\) and \(P(W)=\frac{1}{2}\).
- (iv) Outcomes: "Boy" (B) and "Girl" (G). In the idealised model used in textbooks, \(P(B)=\frac{1}{2}\) and \(P(G)=\frac{1}{2}\).
2. Check whether the two outcomes have the same probability.
- (i) In real life the probability of a car starting is usually greater than the probability of it not starting ( \(p>\frac{1}{2}\) ). Hence the outcomes are not equally likely.
- (ii) For a basketball player the chance of making a shot depends on skill, distance, etc.; generally \(q
eq \frac{1}{2}\). Hence the outcomes are not equally likely.
- (iii) When a true‑false question is attempted by a student who does not know the answer and guesses, each of the two possibilities (right or wrong) has probability \(\frac{1}{2}\). Hence the outcomes are equally likely.
- (iv) Assuming the birth of a boy or a girl is equally probable (the textbook model), each outcome has probability \(\frac{1}{2}\). Hence the outcomes are equally likely.
3. Conclusion: The experiments (iii) and (iv) have equally likely outcomes, while (i) and (ii) do not.
4. Explanation: The definition of equally likely outcomes requires that the sample space consist of outcomes that occur with the same probability. In (i) and (ii) the underlying physical or skill‑related factors make one outcome more probable than the other, violating the condition. In (iii) (random guess) and (iv) (idealised birth of boy/girl) the two outcomes each have probability \(\frac{1}{2}\), satisfying the definition.
Correct Answer: Experiments (iii) and (iv) have equally likely outcomes because each of their two possible results occurs with probability \(\frac{1}{2}\). Experiments (i) and (ii) are not equally likely as the probabilities of the two outcomes are not the same.