When a coin is tossed four times, sixteen equally likely outcomes are possible as shown below: HHHHHHHTHHTHHHTTHTHHHTHTHTTHHTTTTHHHTHHTTHTHTHTTTTHHTTHTTTTHTTTT Let X denote the total number of tails obtained in the four tosses. Find the following probabilities: A. P ( X < 3) = B. P ( X > 2) =
Question
When a coin is tossed four times, sixteen equally likely outcomes are possible as shown below: HHHHHHHTHHTHHHTTHTHHHTHTHTTHHTTTTHHHTHHTTHTHTHTTTTHHTTHTTTTHTTTT Let X denote the total number of tails obtained in the four tosses. Find the following probabilities: A. P ( X < 3) = B. P ( X > 2) =
Solution
A. To find P(X < 3), we need to find the probability of getting 0, 1, or 2 tails in four tosses.
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For 0 tails (all heads): There is only 1 outcome (HHHH), so the probability is 1/16.
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For 1 tail: There are 4 outcomes (THHH, HTHH, HHTH, HHHT), so the probability is 4/16 = 1/4.
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For 2 tails: There are 6 outcomes (TTHH, THTH, THHT, HTTH, HTHH, HHHT), so the probability is 6/16 = 3/8.
Adding these probabilities together, P(X < 3) = 1/16 + 1/4 + 3/8 = 11/16.
B. To find P(X > 2), we need to find the probability of getting 3 or 4 tails in four tosses.
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For 3 tails: There are 4 outcomes (TTTH, TTHT, THTT, HTTT), so the probability is 4/16 = 1/4.
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For 4 tails (all tails): There is only 1 outcome (TTTT), so the probability is 1/16.
Adding these probabilities together, P(X > 2) = 1/4 + 1/16 = 5/16.
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