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1. Blue balls with twice the red probability ●○○○○ 10 ptsA bag contains 6 red balls and some blue balls. The probability of drawing a blue ball from the bag is twice the probability of drawing a red ball. Find the number of blue balls in the bag.
2. Conditional probabilities from given data ●○○○○ 10 ptsGiven $P(E)=0.5$, $P(F)=0.7$ and $P(E \cap F)=0.3$, find $P(E\mid F)$ and $P(F\mid E)$. Round each answer to two decimal places.
3. Estimating red balls from draw frequencies ●○○○○ 10 ptsAn opaque bag contains a total of $20$ balls, each either red or white, identical except for color. A ball is drawn at random, its color is noted, and it is put back; this is repeated many times. The results are: - out of $50$ draws, $11$ red - out of $100$ draws, $27$ red - out of $200$ draws, $50$ red - out of $500$ draws, $124$ red - out of $800$ draws, $201$ red - out of $1000$ draws, $249$ red Estimate the number of red balls in the bag.
4. Union probability from given values ●○○○○ 10 ptsGiven $P(A)=0.6$, $P(B)=0.5$ and $P(A\mid B)=0.3$, find $P(A\cup B)$. Give your answer as a decimal.
5. Die roll greater than two ●○○○○ 10 ptsA fair six-sided die is rolled three times. The results are 1, 4, and 3, in that order. What is the probability that the next roll will produce a number greater than 2? Round to two decimal places.
6. Normal density at a point ●○○○○ 10 ptsFind $P(x)$ for the normal distribution with mean $\mu = 2$ and standard deviation $\sigma = 3$ at $x = 5$, using $$P(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\,e^{-(x-\mu)^2/(2\sigma^2)}.$$ Use $\pi \approx 3.14$. Round to four decimal places.
7. Two heads and two tails ●○○○○ 10 ptsFour fair coins are tossed simultaneously. What is the probability that exactly two coins show heads and two show tails? Give your answer as a decimal.
8. Asteroid collisions in a million years ●○○○○ 10 ptsAsteroids with a diameter of at least 1 km collide with the Earth at a rate of approximately 2 per million years. Assume collisions occur according to a Poisson process. What is the probability that in a randomly selected million-year period there is exactly one collision? Round to four decimal places.
9. Birthday not in December ●○○○○ 10 ptsIgnoring leap years, what is the probability that a randomly chosen person's birthday is not in December? Give your answer as a decimal rounded to three decimal places.
10. Ace then king without replacement ●○○○○ 10 ptsA standard 52-card deck is shuffled. One card is drawn and not replaced, then a second card is drawn. Find the probability that the first card is an ace and the second card is a king. Give your answer as a decimal rounded to four decimal places.

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