Fast Probability and Statistics (No Calculator)

Context: Answer quickly with mental math and concise reasoning. Show set-up and any helpful shortcuts.

1) Expected Value and Variance

A gamble pays +$10 with probability 0.4 and −$6 otherwise. Compute the expected value (EV) and variance (Var).

2) Conditional Probability with Dice

Two fair dice are rolled. Given that the sum is at least 9, what is the probability that at least one die shows a 6?

3) Distribution Choice for Arrivals

Arrivals occur at an average rate of 3 per minute. Choose an appropriate model for the count in 20 seconds and compute P(X ≥ 2).

4) Gambling Odds and Kelly

Decimal odds for event A are 2.6 while the true win probability is 0.35. Compute:

• The implied probability from the odds.
• The edge/expected return per $1 staked.
• The Kelly fraction (optimal fraction of bankroll to bet), assuming a unit bankroll and standard Kelly.

5) Cards Without Replacement

From a standard 52-card deck, what is the probability of drawing at least one Ace in three draws without replacement? Show quick mental-math steps.