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Three different coins are flipped together.
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Find the probability of getting exactly two heads, at least two heads, at least two tails.
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Let’s use systematic listing to first find all possible outcomes when two of the coins are flipped.
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A sample space diagram can be a nice way to do this without losing any of the outcomes.
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The possible outcomes on each coin are heads and tails.
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So this means we could get heads on both coins, heads on the first coin and tails on the second, tails on the first coin and heads on the second, or two tails.
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Now let’s take these outcomes and we’ll add the third coin.
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We could get three heads.
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We could get heads-tails-heads, tails-heads-heads, or tails-tails-heads, and so on, all the way through to the final outcome, which would be three tails.
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Now let’s look at part one.
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It’s asking us to find the probability of getting exactly two heads.
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There are three ways of this happening: heads-tails-heads, tails-heads-heads, and heads-heads-tails.
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Remember, to find the probability of an event occurring, we work out the number of ways that event can occur and we divide it by the total possible number of outcomes.
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There are three ways of getting exactly two heads, and there are a total of eight possible outcomes.
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The probability of throwing exactly two heads is three-eighths.
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Part two is asking us to find the probability of getting at least two heads.
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“At least two heads” means two or more.
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This includes the possible ways of getting exactly two heads plus an additional outcome, and that’s when we get three heads.
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The probability of getting at least two heads is four-eighths, and four-eighths simplifies to one-half.
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So the probability of getting at least two heads is one-half.
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Part three: find the probability of getting at least two tails.
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There are three ways of getting exactly two tails.
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And there’s a fourth possible outcome, and that’s getting three tails.
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Once again, we have the probability of four-eighths, which simplifies to one-half.
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The probability of getting at least two tails is one-half.