A geologist splits rocks to look for fossils. On average $7\%$ of the rocks selected from a partic

Question

A geologist splits rocks to look for fossils. On average $7\%$ of the rocks selected from a particular area contain fossils.

The geologist selects a random sample of $20$ rocks from this area.

Find the probability that at least three of the rocks contain fossils.[2]

A random sample of $n$ rocks is selected from this area.

The geologist wants to have a probability of $0.8$ or greater of finding fossils in at least three of these rocks. Find the least possible value of $n$.[3]

In early 2017, geologists found the fossils of zilantophis schuberti, a new discovered species of winged serpent. On average, the proportion of rocks that contain fossils of zilantophis schuberti in this area is $p$. It is known that the number of rocks containing fossils of zilantophis schuberti in a random sample of $10$ rocks has a unique mode of $3$.

Use this information to find exactly the range of values that $p$ can take.[4]

TGM clarification: the binomial variable counts rocks containing at least one fossil, not individual fossils. In the final part, “mode” means a unique mode, matching the strict inequalities in the retained worked source. Use independent trials with the stated common probability for the binomial model.

Accepted Answer

(i) $0.161$ (3 s.f.). (ii) $n=60$. (iii) No compatible $p$ under the simultaneous same-area premises. The separate source-intended model $K\sim B(10,p)$ gives $\frac3{11}<p<\frac4{11}$ for a unique mode of $3$.

2017 TPJC P2 Q6

2017 TPJC P2 Q6

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