Official Explanation: Remember that, to calculate the probability of an "at least" scenario, we use the complement rule. The case we want is (at least one H in two flips). The complement of that is (no H in two flips). If p is the probability of H, then (1 – p) is the probability of T. The probability of two T in two flips would be that squared, and then we would subtract from 1 to find the "at least" probability. In the table below, the first column is the possible values of p, the probability of getting H on a single flip. The second column is the probability of getting T on a single flip. The third column is the probability of getting two T's in a row, i.e. no H in two flips; that is the complement of the "at least" case. The final column is the probability of "at least one H in two flips."
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We see that for all value of p ≥ 0.3, the "at least" probability is greater than 0.5.
FAQ: Why is the (1 - p) term being squared?We know that:
P(heads) = p
P(tails) = (1 - p)
This problem ultimately asks us to find the probability of getting "at least one heads in two flips". This means that we want to find the probability of getting the following outcomes:
heads, heads
OR
heads, tails
OR
tails, heads
We can calculate all this more easily by first finding the complement to that. The complement to getting "at least one heads in two flips" is getting "exactly 2 tails":
tails, tails
Thus, we're looking for the probability of getting tails AND tails:
P(exactly 2 tails) = P(tails) * P(tails) =
P(tails)^2Substituting in for the value of P(tails), we get:
P(exactly 2 tails) = (1 - p)^2
Taking the complement of this gives us the final expression for our chart:
P(at least one heads) = 1 - P(exactly 2 tails)
P(at least one heads) = 1 - (1 - p)^2
FAQ: Do we really have to make that whole chart? That would take too long!No, you don't have to fill in that whole chart. That's just being used to illustrate the thinking behind this problem. To solve the problem, you really only need to know that expression in the last column: 1 - (1 - p)^2. We know that that expression must be greater than 0.5. So we end up with:
1 - (1 - p)^2 > 0.5
Now you can just plug in the different answer choices as the value for p in this expression and see which values yield a true statement.
01 Dec 2018, 10:54
40% (02:17) correct
