Exact probabilities, cumulative probabilities and model assumptions.
Example 1
X counts successes in 7 independent trials, each with success probability 0.8. Find P(X=4). Give your answer to 4 decimal places.
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Answer: 0.1147
- X is binomial with n=7,p=0.8.
- P(X=4)=C(7,4)×0.84×0.23.
- Rounded answer: 0.1147.
Example 2
X is binomial with n=8,p=0.7. Find P(X≥3). Give your answer to 4 decimal places.
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Answer: 0.9887
- Use the complement: P(X≥3)=1−P(X≤2).
- Use the binomial cumulative distribution, including values 0 through 2.
- Rounded answer: 0.9887.
Example 3
A bag has 5 red and 7 blue counters. 3 counters are drawn with replacement, mixing well before each draw. Can the number of red counters be modelled exactly by a binomial distribution?
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Answer: Yes: independent draws with constant success probability
- A binomial model needs a fixed number of independent trials with constant success probability.
- Replacing and mixing keeps P(red)=125 on every draw.