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# expected value

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Suppose you have a weighted coin in which heads comes up with probability 3/4 and tails with probability 1/4. If you flip heads, you win $1 but if you flip tails, you lose$1 What is the expected win of a coin flip in dollars?

Jul 27, 2022

#1
+1

Suppose you have a weighted coin in which heads comes up with probability 3/4 and tails with probability 1/4. If you flip heads, you win $1 but if you flip tails, you lose$1 What is the expected win of a coin flip in dollars?

I don't know the formal way to solve this kind of problem, so I'll just kind of think out loud.

The first three times you flip the coin, it's heads so you win $3. The fourth time you flip the coin, it's tails so you lose$1.

You've won $3 and lost$1 so you've netted $2. You've netted$2 and it took four flips to do it, so that comes to \$0.50 per filp.

You can reflect on whether or not this reasoning is legitimate while you wait for one of the smart ones to come along.

Jul 28, 2022
#2
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The expected value is $$({3 \over 4} \times 1) + (-1 \times {1 \over 4}) = \color{brown}\boxed{1 \over 2}$$

Jul 28, 2022