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Unformatted text preview: Bio 151 Investigations in Genetics and Evolution Prof. Autumn Binomial probabilities Imagine that you and your new roommate begin to argue over who will get the top bunk on the bunk bed. Your roommate suggests that you flip a coin to decide fairly. As soon as you agree, your roommate produces a wellworn antique coin of unknown origin, quickly flips it, calls out “Heads!”, and catches the coin. The result is heads. The roommate grins and begins to climb to the top bunk. You say, “Wait! How do I know that the coin is fair?” The roommate tosses you the coin and says, “Here take a look.” What will you do? How could you determine if the coin was fair? Let’s define “fair” to mean that there are equal probabilities of the coin landing with heads or tails facing up. We will call the hypothesis that the coin is fair the null hypothesis , or H . The hypothesis that the coin is not fair is the alternative hypothesis, or H a . In mathematical terms, p = probability of heads q = probability of tails p + q = 1 (by definition) If the null hypothesis is true, and you flip the coin once, there’s a 0.5 probability of getting a head and a 0.5 probability of getting a tail. If the null hypothesis is false (H a is true), there would be a greater probability of getting a head or a tail. How can you test H ? If you flip the coin repeatedly and always get heads, then H seems unlikely to be true. So, you flip the coin 5 times, and this is what happens: {H, H, H, T, H} “Hey!” you say, “This coin flips heads 80% of the time!” Your roommate retorts, “No way!”, takes the coin and flips it 5 times. This is what happens: {T, H, T, H, H} “Way!” you say. “You still flipped 60% heads and only 40% tails.” How can you quantitatively address the question of whether on not this is a fair coin? Given your observations of the coin, how likely is it that the null hypothesis (H ) is true? What are the chances of getting only 1 head in 5 flips if the coin is really fair (if H is really true)? Let’s begin by calculating the chances of getting 1 or fewer heads out of 5 flips if the coin is fair. If this probability is low, you can conclude that it’s not likely that the coin is fair (i.e. H is probably false and can be rejected)....
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 Spring '10
 Autumn
 Genetics, Evolution, Null hypothesis, Probability theory, Coin flipping, Coin, Prof. Autumn

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