The probability that Kersley is either asleep or studying at noon on any given day is the sum of the probabilities of these two independent events:
\(P(A) = 0.2 + 0.6 = 0.8\).
We need to find the probability that Kersley is either asleep or studying on at least 6 days out of 7. This is a binomial probability problem with parameters \(n = 7\) and \(p = 0.8\).
The probability of Kersley being asleep or studying on exactly 6 days is given by:
\(P(6) = \binom{7}{6} (0.8)^6 (0.2)^1\).
The probability of Kersley being asleep or studying on all 7 days is:
\(P(7) = (0.8)^7\).
Thus, the probability that Kersley is either asleep or studying on at least 6 days is:
\(P(6 \text{ or } 7) = P(6) + P(7)\).
Calculating these:
\(P(6) = \binom{7}{6} (0.8)^6 (0.2)^1 = 7 \times 0.262144 \times 0.2 = 0.3670016\).
\(P(7) = (0.8)^7 = 0.2097152\).
Therefore,
\(P(6 \text{ or } 7) = 0.3670016 + 0.2097152 = 0.5767168\).
Rounding to three decimal places, the probability is \(0.577\).
