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In this question, we consider a fair coin that is tossed four times, and the random variable x is the number of heads in four tosses.
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For part a, we are asked to find the probability distribution of x.
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Now, because it's a fair coin, each time we, each of these coins has a probability of being heads of one half.
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So let's call ahead a success, so one half.
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And we're tossing the coin four times.
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So here each of these tosses can be viewed as a bernoulli trial with two outcomes of interest.
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It's either heads or not.
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And it's a reason to believe that their outcomes are independent.
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And so the number of successes and a fixed number of independent brunuli trials is a binomial random variable.
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So here, x is a binomial based on four trials and probability of success .5.
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The probability function for the binomial random variable is given by this formula.
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And so to make the probability distribution, we just have to find the probabilities for each possible value of x.
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So x can be any integer from zero up to four.
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So for x equals zero, the probability function simplifies to one -half to the exponent four.
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Actually, before we do this, let's specify the probability function for this particular set of parameters.
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So the probability function, when we have four trials and probability of success, one half, probability function simplifies to 4 choose x times 1 half to the exponent 4.
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And that's for x is any integer from 0 up to 4.
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So for x equal to 0, probability comes out to 1 over 16.
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For x equals 1 comes out to 1 over 4, or x equals 2 comes out to 0 .375.
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And for x equals 3, we have 1 quarter again, and for x equals 4, 1 over 16.
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This can be done quite a bit quicker using software such as excel.
02:54
So in excel, if we, in column a, put the different possible values for x, and then in column b, we enter a formula, we start with equals...