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Solution: 2016 Fall Final - 21

Author: Michiel Smid

Question

Consider a coin that comes up heads with probability 1/7 and comes up tails with probability 6/7. You flip this coin once. If it comes up heads, you win \$5. If it comes up tails, you win \$2.
What is the expected value $\mathbb{E}(X)$ of $X$?
(a)
17/7
(b)
7/17
(c)
7/2
(d)
32/7

Solution

  • Let's determine X=5
    The probability of getting heads is $ \frac{1}{7} $
    $ Pr(X=5) = \frac{1}{7} $
  • Let's determine X=2
    The probability of getting tails is $ \frac{6}{7} $
    $ Pr(X=2) = \frac{6}{7} $

$ \mathbb{E}(X) = 5 \cdot Pr(X=5) + 2 \cdot Pr(X=2) $

$ \mathbb{E}(X) = 5 \cdot \frac{1}{7} + 2 \cdot \frac{6}{7} $

$ \mathbb{E}(X) = \frac{5}{7} + \frac{12}{7} $

$ \mathbb{E}(X) = \frac{17}{7} $