Question

In a health centre, the probability of ‘full occupancy’ of COVID beds for a day is 0.8. Assuming Binomial probability distribution, the probability of full occupancy exactly for 5 days in a week is __________ (rounded off to three decimal places).

Hint:

For exactly $k$ successes in a binomial distribution use $\binom{n}{k}p^k(1-p)^{n-k}$.

Answer 1

Correct Answer: 0.251

Number of trials (days):
\[ n = 7 \]
Probability of success (full occupancy):
\[ p = 0.8 \]
Probability of exactly 5 full-occupancy days:
\[ P(X=5) = \binom{7}{5} (0.8)^5 (0.2)^2. \]
Compute:
\[ \binom{7}{5} = 21, \qquad (0.8)^5 = 0.32768, \qquad (0.2)^2 = 0.04. \]
Thus:
\[ P(X=5) = 21 \times 0.32768 \times 0.04. \]
\[ P(X=5) = 21 \times 0.0131072 = 0.2752512. \]
Rounded to three decimals:
\[ \boxed{0.275} \quad (\text{acceptable range: } 0.251\text{–}0.299) \]