Answers: Probability Fundamentals

I strongly recommend trying out these problems thoroughly before checking the answers. You may obtain the same answer through different reasoning, so be sure to check if they are equivalent even if it is written differently than you may have approached! (Even better, see if you can get the same answer through multiple different approaches!)

Essential Problems

  1. $P(\text{Two or more vowels in the chosen word}) =\frac{5}{11} = 0.4545\ldots$
    • (a) $P(\text{Both results } \leq 4) = \left(\frac{4}{6}\right)^2 = 0.444\ldots$
    • (b) $P(\text{minimum } \leq 4) = 1 - \left( \frac{2}{6}\right)^2 = 0.888 \ldots$
  3. $P(\text{not all the same}) = 1- (\frac{1}{2})^2 = \frac{3}{4}$
    • (a) $P(\text{At least 1 lefty}) = 1 - 0.9^5 \approx 0.410$
    • (b) $P(\text{All the same handedness}) = 0.9^5 + 0.1^5 \approx 0.590$
  5. $P(\text{no cat, no dog}) = 0.42$

More challenging problems

1.

  • (a) $P(\text{Sum } = 2) = \frac{1}{36} = 0.0277\ldots$
  • (b) $P(\text{Sum } = 7) = \frac{6}{36} = 0.166\ldots$
    • (a) $P(A\cap(BC)^c) = P(A) - P(AB) -P(AC) + P(ABC)$
    • (b) $P(\text{Exactly two occur}) = P(AB) +P(BC) +P(AC) - 3P(ABC)$
    • (c) $P(\text{Exactly one occurs}) = P(A)+ P(B) + P(C) -2P(AB) -2P(BC) -2P(AC) - 3P(ABC)$

  2. $P(\text{d8 } \geq \text{ d6}) = \frac{33}{48} = 0.6875$

4.

  • (a) $P(\text{No ones OR no twos}) = \left(\frac{5}{6}\right)^5 + \left(\frac{5}{6}\right)^5 - \left(\frac{4}{6}\right)^5 \approx 0.672$
  • (b) $P(\text{At least one 1 and at least one 2} = 1 - \left(\frac{4}{6}\right)^5 \approx 0.868$
  • (c) $P(\text{Only ones and twos}) = \left(\frac{2}{6}\right)^5 \approx 0.0041$
  • (d) $P\text{Both ones and twos, and no other faces}) = \left(\frac{2}{6}\right)^5 - 2\left(\frac{1}{6}\right)^5 \approx 0.00386$