Lauren is the point guard on her basketball team. She estimates that her chances of gettibg a 3 poibter are 85% when there is bo defending against her, but only 60% when there is a defender. There is a 50% chance that the girl assigned to defend her will not be close enough to block the shot. What is the probability of Lauren scoring a 3 point shot?
To calculate the probability of Lauren scoring a 3-point shot, we need to consider two scenarios:
1. The defender is not close enough to block the shot (50% chance).
2. The defender is close enough to block the shot (50% chance).
1. Scenario: The defender is not close enough to block the shot.
In this case, Lauren's chances of getting a 3-pointer are 85%, as there is no defending against her. Thus, the probability of Lauren scoring in this scenario is 85% or 0.85.
2. Scenario: The defender is close enough to block the shot.
In this scenario, Lauren's chances of getting a 3-pointer decrease to 60%. Since the probability of the defender being close enough to block the shot is 50%, the overall probability in this scenario is 60%*50% = 30% or 0.30.
Now, let's calculate the total probability:
Total probability = Probability of scenario 1 + Probability of scenario 2
Total probability = 0.85 + 0.30
Total probability = 1.15 or 115%
However, probabilities must always be between 0 and 1. Based on this information, it seems that the given probabilities may be incorrect, as the total probability exceeds 100%.
To find the probability of Lauren scoring a 3-point shot, we need to consider two scenarios: when there is no defending against her and when there is a defender.
Scenario 1: No defending against her.
The probability of Lauren scoring a 3-point shot without any defending is 85% (0.85).
Scenario 2: Defending against her.
In this scenario, there is a 50% chance that the defender will not be close enough to block the shot. Let's call this event A. The probability of A occurring is 50% (0.5).
If the defender is not close enough to block the shot (event A), Lauren's chances of scoring a 3-point shot drop to 60% (0.6).
Putting it all together, we can calculate the overall probability using the law of total probability:
Probability = (Probability of no defense against her) + (Probability of defense and no block) × (Probability of scoring)
Probability = 0.85 + (0.5 × 0.6)
Probability = 0.85 + 0.3
Probability = 1.15
However, probabilities cannot exceed 1, so the probability of Lauren scoring a 3-point shot is 1 (100%).