Practice section 1 of 7

For each experiment, determine whether X is binomial and state its parameters where appropriate.

(a) Draw a bead ten times without replacement from a jar containing 15 red and 25 blue beads; X counts red beads obtained.

(b) Repeat with the same 15 red and 25 blue beads, but with replacement on every draw.

(c) Randomly answer 20 true/false questions; X counts correct answers.

(d) Randomly answer 20 multiple-choice questions, some with four choices and the rest with five; X counts correct answers.

In a particular large population, 15% of people are left-handed and the rest are right-handed. Twelve people are selected at random. Find the probability that (i) exactly 2 are left-handed, (ii) at most 3 are left-handed, (iii) more than 4 are left-handed, (iv) at least 7 are not left-handed, and (v) there are between 3 and 9 left-handed people.

For $X\sim B(23,0.34)$, evaluate (i) $P(X=4)$, (ii) $P(X\le6)$, (iii) $P(X<5)$, (iv) $P(X>4)$, (v) $P(X\ge14.5)$, (vi) $P(7<X\le16)$, and (vii) $P(\lvert X-8\rvert<5.6)$.

A factory packs chocolates into boxes of 20. A chocolate fails the minimum packing criterion if it weighs less than 20 grams. On average 2% of chocolates fail. (i) Find the probability that a randomly chosen box contains at least one failing chocolate. (ii) For four randomly chosen boxes, find the probability that exactly two contain at least one failing chocolate.

On average 1 in 100 light bulbs made by a factory is faulty. A sample contains n light bulbs and X counts the faulty bulbs. (i) State in context two assumptions needed for a binomial model. (ii) Require $P(X=0)>0.5$. Find the largest possible n.

Twenty percent of batteries produced by a machine are faulty. Find the smallest sample size for which the probability of fewer than 12 faulty batteries is less than 0.65.