A restaurant serves a signature dish. The probability that a diner orders the dish is $0.7$. It ma

Question

A restaurant serves a signature dish. The probability that a diner orders the dish is $0.7$. It may be assumed that each diner orders at most one portion of the dish and the number of diners who order the dish follows a binomial distribution.

Find the probability that $10$ diners order at least $3$ portions of the signature dish.[2]
Find the probability that $10$ diners order between $3$ and $8$ portions of the signature dish.[2]
If the restaurant has enough ingredients to prepare $80$ portions of the signature dish, find the maximum possible number of diners if the restaurant has at least $90%$ chance of meeting the number of orders for the dish.[2]

On a particular day, the restaurant caters for a function with $40$ tables of $10$ diners each.

Find the probability that each table orders at least $3$ portions of the signature dish.[1]
Find an estimate of the probability that the average number of portions of the signature dish served per table is at least $6.9$.[3]

Accepted Answer

(a) ${\mathrm{P}(X \geq 3)=0.998}$ ($3$ s.f.) (b) ${\mathrm{P}(3 < X < 8)=0.607}$ ($3$ s.f.) (c) Largest ${n=106}$ (maximum number of diners) (d) ${0.938}$ ($3$ s.f.) (e) ${\mathrm{P}\left(\overline{X}\geq 6.9\right)=0.669}$ ($3$ s.f.)

2024 NYJC P2 Q9

2024 NYJC P2 Q9

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