Suppose an unbiased coin is tossed 6 times. Each coin toss is independent of…
2026
Suppose an unbiased coin is tossed 6 times. Each coin toss is independent of all previous coin tosses. Let 𝐸1 be the event that among the second, fourth, and sixth coin tosses, there are at least two heads. Let 𝐸2 be the event that among the first, second, third, and fifth coin tosses, there are equal number of heads and tails. The conditional probability P(𝐸1 | 𝐸2) is equal to ____________. (rounded off to one decimal place)
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Correct answer: 0.5
Let the tosses be numbered 1 to 6. Event E2 says that among tosses 1, 2, 3 and 5 there are exactly two heads.
By symmetry under E2, toss 2 is still equally likely to be H or T, so P(toss 2 is H | E2) = 1/2. Tosses 4 and 6 are independent of E2 and remain fair coin tosses.
Therefore, under E2, the three tosses relevant to E1, namely tosses 2, 4 and 6, behave like three independent fair tosses. The probability of at least two heads among three fair tosses is:
C(3,2)/2^3 + C(3,3)/2^3 = 3/8 + 1/8 = 4/8 = 0.5.
Thus P(E1 | E2) = 0.5, rounded to one decimal place.