Tensions are high between the Saxons and the Vikings, and King Magnus of the Vik

Question

Tensions are high between the Saxons and the Vikings, and King Magnus of the Vikings fears an invasion. He decides to deploy 50 elite soldiers to set up an ambush camp near where the Saxons are positioned. These 50 elite soldiers must complete an intense week of training before deployment; so intense, that each soldier has a 4% chance of sustaining an injury. In preparation for this, King Magnus decides to nominate 5 reserve soldiers, who will be randomly selected to replace any elite soldier injured during the training. These are the only 55 men that King Magnus can dedicate to this operation. One of the reserve soldiers is called Mata. (a) What is the probability that King Magnus has not nominated enough reserve soldiers? (b) What is the probability that Mata will be required to join the ambush camp?

Explanation / Answer

Solution:

(a)

King has not nominated enough reserve soldiers Number of soldiers injured is more than 5. Since the King has provided for only 5 reserve soldiers, there would no reserve soldier to replace the injured.

Now, let X = Number of soldiers injured during training period. Since there are 50 soldiers in the training camp and 4% get injured (which is the same as probability of a soldier getting injured is 0.04), X ~ B (50, 0.04).

So, King has not nominated enough reserve soldiers X > 5 and hence the required probability

= P(X > 5) = 1- P(X 5) = 1- 0.98559 = 0.01441

(b)

Since Mata is one among the 5 reserve soldiers and soldiers are allotted to replace injured ones at random, probability Mata will be deployed is 1/5, provided one or more soldiers under training get injured.

Now, P(one or more soldiers under training get injured) = 1 – P(None gets injured)

= 1 – P(X = 0) = 1 – 0.1299 = 0.8701

So, P (Mata will be required to join the ambush camp) = (1/5) × 0.8701 = 0.1740

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