Fighting probability of fallow deer bucks. In Aggressive Behavior (Jan./Feb.2007

Question

Fighting probability of fallow deer bucks. In Aggressive Behavior (Jan./Feb.2007), zoologists investigated the likelihood of fallow deer bucks fighting during mating season. During a 270-hour observation period, the researchers recorded 205 encounters between two bucks. Of these, 167 involved one buck clearly initiating the encounter with the other. In these 167 initiated encounters, the zoologists kept track of whether or not a physical contact fight occurred and whether the initiator ultimately won or lost the encounter. (The buck that is driven away by the other is considered the loser.) A summary of the 167 initiated encounters is provided in the accompanying table. Suppose we select one of these 167 encounters and note the outcome (fight status and winner).

(Give answers to three decimal places.)

a) Given that a fight occurs, what is the probability that the initiator wins?  

b) Given no fight, what is the probability that the initiator wins?  

c) Are the events "no fight" and "initiator wins" independent? Explain.

Initiator Wins No Clear Winner Initiator Loses Totals Fight 26 23 15 64 No Fight 80 12 11 103 Totals 106 35 26 167

Explanation / Answer

Solution:-

a) Given that a fight occurs, the probability that the initiator wins is 0.4063.

Number of times fight occurs = 64

Number of times the initiator wins = 26

Given that a fight occurs, the probability that the initiator wins = 26/64 = 0.4063.

b) Given no fight, the probability that the initiator wins is 0.7767

Number of times no fight occurs = 103

Number of times the initiator wins = 80

Given no fight, the probability that the initiator wins = 80/103 = 0.7767

c) No the events "no fight" and "initiator wins" are not independent.

P(A And B) = P(A) × P(B)

P(A And B) = 80/167

P(A And B) = 0.479

P(A) = 103/167 = 0.6168

P(B) = 106/167 = 0.6347

P(A) × P(B) = 0.6168 × 0.6347

P(A) × P(B) = 0.3915

Since P(A And B) is not equal to P(A) × P(B), hence the events "no fight" and "initiator wins" are not independent

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