Binomial Distribution Math Example 4

Follow the full solution, then compare it with the other examples linked below.

medium

A student guesses on

7

true-or-false questions. What is the probability of getting at least

6

correct?

1
Let $X \sim \text{Binomial}(7, 0.5)$ . We want $P(X \ge 6) = P(X = 6) + P(X = 7)$ .
2
$P(X = 6) = \binom{7}{6}(0.5)^6(0.5)^1 = 7(0.5)^7$ .
3
$P(X = 7) = \binom{7}{7}(0.5)^7 = (0.5)^7$ .
4
Add them: $7(0.5)^7 + (0.5)^7 = 8(0.5)^7 = \frac{8}{128} = \frac{1}{16}$ .

Answer

P(X \ge 6) = \frac{1}{16} = 0.0625

Binomial probabilities add cleanly when several exact-success counts are acceptable. Here, 'at least

6

' means combining the probabilities for exactly

6

and exactly

7

successes.

About Binomial Distribution

The probability distribution of the number of successes in $n$ independent yes/no trials, each with probability $p$ .

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