Digital SATMath: Problem-Solving and Data AnalysisHard
A meteorologist is tracking rainfall. The probability of rain on any given day is 0.4. What is the probability that it will rain on exactly 2 out of 3 consecutive days, assuming the days are independent?
- A0.480
- B0.288
- C0.576
- D0.192
Show answer & explanationAnswer & explanation
Correct answer: B. 0.288
Let R be rain (P(R)=0.4) and N be no rain (P(N)=1-0.4=0.6). We want exactly 2 rain days out of 3. The possible sequences are RRN, RNR, NRR. Probability of each sequence: P(RRN) = 0.4 * 0.4 * 0.6 = 0.096. P(RNR) = 0.4 * 0.6 * 0.4 = 0.096. P(NRR) = 0.6 * 0.4 * 0.4 = 0.096. Total probability = 0.096 + 0.096 + 0.096 = 3 * 0.096 = 0.288.
Why the other options are wrong
- A. This might result from a miscalculation, such as multiplying 0.4 * 0.6 * 2 or similar error.
- C. This might come from an incorrect number of permutations (e.g., 6 instead of 3) or a calculation error.
- D. This is the probability of one specific sequence (e.g., RRN) but not accounting for all permutations.
Binomial Probability
Binomial probability calculates the likelihood of getting exactly 'k' successes in 'n' independent Bernoulli trials, where each trial has only two possible outcomes (success/failure).
- Requires fixed number of trials (n).
- Each trial has only two outcomes (success/failure).
- Probability of success (p) is constant for each trial.
- Trials are independent.
Memory trick: How many ways to get successes, times probability of one way.