The given problem involves the UF team trailing by two points with only 8 seconds to go. The first option is to go for a 2-point shot which will result in a tie and the game goes overtime. The probability of winning a 2-point is 45% and the probability of winning in overtime is 50%. Therefore, the overall chance of winning with a 2-point shot is given by multiplying both probabilities as follows:
= P (win a 2-point) * P (win in overtime)
= 0.45 * 0.55
= 22.5%
The second option involves going for a 3-point which will result in a win and the game will not go into overtime. The probability of winning a 3-point shot is 30%.
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From the analysis of the given probabilities, the 3-point shot has a higher probability of 30% while the 2-point shot has a lower probability of 22.5%. Therefore, the team should go for the 3-point shot as it has a higher probability.
The other factor that should be considered in the game is the chances of winning in overtime. When the 2-point shot was considered alone, it had a higher probability of 45% as compared to 30% for the 3-point. However, this did not provide an accurate overview of the overall probability. The 2-point would result in overtime which would further impact the probabilities of winning.
Coach's Decision: University of Washington vs. University of Notre Dame match
The problem involves Washington vs. University of Notre Dame match with 3.5 minutes left in the game. Washington had a fourth-down, one yard to go for a touchdown, and led by two points. Notre Dame had prevented Washington’s touchdown twice from the same position. Washington’s coach decided not to go for a touchdown and this led to the team losing. The decision was analyzed in a decision tree to determine whether the team made the best decision. The results for the option of going for a field goal were as follows:
Going for a field goal
= P (success of field goal) * P (Notre Dame’s offense don’t score)
= 90% * 50% = 45%
= P (success of field goal) * P (Notre Dame’s score 6 points) * P (Washington win last min.)
= 90% * 50%* 20% = 9%
= P (success of field goal) * P (Notre Dame’s score 8 points) * P (Washington win last min.)
= 90% * 0%* 5% = 0%
= P (fail at field goal) * P (Notre Dame’s offense don’t score)
= 10% * 40% = 4%
= P (fail at field goal) * P (Notre Dame’s get field goal) * P (Washington win last min.)
= 10% * 30%* 20% = 0.6%
= P (fail at field goal) * P (Notre Dame’s get touchdown) * P (Washington win last min.)
= 10% * 30%* 10% = 0.3%
Total probability of Washington winning when going for a field goal
= 45% + 9% + 4% +0.6% + 0.3% = 58.9%
The overall probability of Washington winning in case they went for a field goal was calculated as 58.9%. The overall probability of Washington winning in case they went for a touchdown was given as 73.75%. The touchdown option had a higher probability of winning compared to the field goal option. The coach thus made the wrong decision by choosing to go for a field goal which had a lower chance of winning.
