Two fair six-sided dice are rolled. What is the probability that the sum of the numbers shown on the top faces is equal to 8?
Total outcomes = 6 * 6 = 36. Favorable outcomes for sum 8: (2,6), (3,5), (4,4), (5,3), (6,2) -> 5 outcomes. Probability = 5/36.
A fair coin is tossed 4 times. What is the probability of getting at least 3 heads? (Express as a decimal rounded to 4 decimal places)
Total outcomes = $2^4$ = 16. Outcomes with 3 heads = C(4,3) = 4. Outcomes with 4 heads = C(4,4) = 1. Total favorable = 5. Probability = 5/16 = 0.3125.
If P(A) = 0.4, P(B) = 0.5, and P(A U B) = 0.7, what is P(A | B)?
$P(A U B) = P(A) + P(B) - P(A ∩ B) => 0.7 = 0.4 + 0.5 - P(A ∩ B) => P(A ∩ B) = 0.2. P(A | B) = P(A ∩ B) / P(B) = 0.2 / 0.5 = 0.4.$
Events A and B are independent. If P(A) = 0.3 and P(B) = 0.6, what is P(A U B)?
For independent events, P(A ∩ B) = P(A) * P(B) = 0.3 * 0.6 = 0.18. P(A U B) = P(A) + P(B) - P(A ∩ B) = 0.3 + 0.6 - 0.18 = 0.72.
An urn contains 5 red balls and 7 blue balls. Two balls are drawn at random one after another without replacement. What is the probability that both balls are red? (Express as a fraction in lowest terms or decimal rounded to 4 decimal places)
$P(1st red) = 5/12. P(2nd red | 1st red) = 4/11. P(both red) = (5/12) * (4/11) = 20/132 = 5/33 \approx 0.1515.$