A football team is known to run 30% of its plays to the left and 70% to the right. A linebacker on an opposing team notices that, when plays go to the right, the right guard shifts his stance most of the time (80%) and that he uses a balanced stance the remainder of the time. When plays go to the left, the guard takes a balanced stance 90% of the time and the shift stance the remaining 10%. On a particular play, the linebacker notes that the guard takes a balanced stance. (a) What is the probability that the play will go to the left

0.6585

Step-by-step explanation:

Given the following :

Proportion of play to the left = 30% = 0.3

Proportion of play to the right = 70% = 0.7

Play to the right:

Shift stance = 80% = 0.8

Balanced stance = (1 - 0.8) = 0.2

Play to the right with balanced stance = (0.2 × 0.7) = 0.14

Play to the right with shifted stance = (0.8 × 0.7) = 0.56

Play to the left:

Balanced stance = 90% = 0.9

Shifted stance = (1 - 0.9) = 0.1

Play to the left with balanced stance = (0.9 × 0.3) = 0.27

Play to the left with shifted stance = (0.9 × 0.3) = 0.03

Probability that play will go to the left if the linebacker takes a balanced stance :

Overall proportion of play with balanced stance :

Balanced with play to the right + balanced with play to the left = (0.14 + 0.27) = 0.41

Balanced play to the left = 0.27

= (balanced play to the left / total proportion of balanced stance)

= 0.27 / 0.41

= 0.6585365

= 0.6585