Question

You have to decide, giving a reason, whether the binomial distribution is a suitable model or not.

(a) Modelling the number of throws of a die untila six is thrown.
(b) Modelling of picking different numbers of red balls, when the balls being picked are not put back into the bag. Initially there are 6 red and 4 black balls in the bag.
(c) Modelling the numbers of faulty light bulbs in a batch of 20 bulbs if it is known that the probability of selecting a faulty bulb is 0.03.
(d) Modelling the number of times the bull is hit in a game of darts when 40 darts are thrown at the bull, if the probability of hitting the bull is 0.12.
(e) Modelling the number of breakages in a batch of 50 wine glasses being delivered by a courier.
(f) Modelling the number of snowdrop bulbs flowering out of 15 bulbs planted if the individual probability that a bulb flowers is 0.7.

Answer 1

The binomial distribution is not suitable for (a) and (b), but it is suitable for (c).

Step-by-step explanation:

(a) The binomial distribution is not a suitable model for modelling the number of throws of a die until a six is thrown. This is because each throw of a die is an independent event with two possible outcomes (success: rolling a six or failure: rolling any other number), and the probability of success (rolling a six) does not remain the same for each trial. Therefore, a geometric distribution is a more appropriate model for this scenario.

(b) The binomial distribution is not a suitable model for modelling the picking of different numbers of red balls without replacement from a bag initially containing 6 red balls and 4 black balls. In a binomial distribution, the trials are independent and the items are replaced after each trial. However, in this scenario, the trials are not independent as the items are not replaced after each pick. Therefore, a hypergeometric distribution is a more suitable model.

(c) The binomial distribution is a suitable model for modelling the numbers of faulty light bulbs in a batch of 20 bulbs, given that the probability of selecting a faulty bulb is known to be 0.03. In this case, each bulb is treated as an independent trial with two possible outcomes (success: selecting a faulty bulb or failure: selecting a non-faulty bulb), and the probability of success (selecting a faulty bulb) remains the same for each trial.