Question at position 10 You play tennis regularly with a friend, and from past experience, you believe that the outcome of each match is independent. For any given match you have a probability of 0.4 of winning. What is the probability that you win at least one of the next five matches?0.01020.98982.00000.07780.9222

Question

BlackTom AI · Accepted Answer

Step into the scenario: you have five independent tennis matches, each with probability p = 0.4 of winning. To find the chance of at least one win, it's often easier to calculate the complement: no wins in all five matches, and subtract from 1.

Option A: 0.0102
This would correspond to a probability approaching 1% of something occurring. It does not match the complement calculation here. The probability of zero wins in five matches is (1-0.4)^5 = 0.6^5 = 0.07776, which is about 7.776%, not 1.02%.

Option B: 0.9898
This is very close to 1, and it actually represents 1 minus 0.0102. Since our computed at-least-one-win probability is 1 − 0.07776 ≈ 0.92224, 0.9898 would be far too high. It would imply a much smaller chance of no wins than correct here.

Option C: 2.0000
This value is outside the range [0,1] for a probability and is therefore invalid in the context of a probability of an event. It cannot be correct.

Option D: 0.0778
This number is the probability of getting no wins in all five matches, i.e., (0.6)^5 ≈ 0.07776. It represents the complement piece, not the probability of at least one win. It is the probability of the complementary event, not the event of interest.

Option E: 0.9222
This matches the calculated probability of at least one win: 1 − (0.6)^5 ≈ 1 − 0.07776 = 0.92224, which rounds to 0.9222. It correctly captures the likelihood of achieving at least one victory over five independent matches with p = 0.4.

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