With the Summer Olympics scheduled to be held this year (hopefully!), those in charge of choosing teams for the various athletic teams (swimming, gymnastics, track and field, etc.) in which their countries will be competing must first need to decide which athletes that they will be sending to compete in Japan. Once the athletes are chosen, the next decisions that must be made involve selecting which specific athletes will be chosen to complete in specific events, especially the team events such as the swimming relays. Suppose that the coaches in change of the swimming events for Canada need to select which of the four swimmers that they will select to compete in the 4 by 400 metre men’s relay. Canada only has 6 swimmers that are good enough to compete in one or more of the four strokes (backstroke, breaststroke, butterfly and free style) that make up this relay. The job of these coaches is to decide which of the 4 swimmers they will choose and who they will choose to compete in which stroke. The coaches have decided to choose which swimmer to compete in each particular stroke based on each swimmer’s average time in each of his last 5 times in these particular strokes. The following table lists these average times for each swimmer in each of these strokes. Times in minutes: Stoke Swimmer breaststroke backstroke butterfly freestyle George 4.25 5.25 4.15 3.85 Tyler 4.52 5.05 4.35 3.98 Ahmed 4.37 5.13 4.45 4.07 John 4.62 5.21 4.53 3.99 Gustov 4.38 5.17 4.58 4.12 Kevin 5.03 5.01 4.53 4.01 Using xij’s as was used in our discussion in the classroom and in our examples, formulate the following to enable the coaches to choose who should compete in each event based on the above average times. the objective function (only include the first 2 entries, ‘…’ and the last two entries) the basic constraint for George the basic constraint for the backstroke
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
With the Summer Olympics scheduled to be held this year (hopefully!), those in charge of choosing teams for the various athletic teams (swimming, gymnastics, track and field, etc.) in which their countries will be competing must first need to decide which athletes that they will be sending to compete in Japan. Once the athletes are chosen, the next decisions that must be made involve selecting which specific athletes will be chosen to complete in specific events, especially the team events such as the swimming relays. Suppose that the coaches in change of the swimming events for Canada need to select which of the four swimmers that they will select to compete in the 4 by 400 metre men’s relay. Canada only has 6 swimmers that are good enough to compete in one or more of the four strokes (backstroke, breaststroke, butterfly and free style) that make up this relay. The job of these coaches is to decide which of the 4 swimmers they will choose and who they will choose to compete in which stroke. The coaches have decided to choose which swimmer to compete in each particular stroke based on each swimmer’s average time in each of his last 5 times in these particular strokes. The following table lists these average times for each swimmer in each of these strokes.
Times in minutes: Stoke
Swimmer breaststroke backstroke butterfly freestyle
George 4.25 5.25 4.15 3.85
Tyler 4.52 5.05 4.35 3.98
Ahmed 4.37 5.13 4.45 4.07
John 4.62 5.21 4.53 3.99
Gustov 4.38 5.17 4.58 4.12
Kevin 5.03 5.01 4.53 4.01
Using xij’s as was used in our discussion in the classroom and in our examples, formulate the following to enable the coaches to choose who should compete in each
- the objective
function (only include the first 2 entries, ‘…’ and the last two entries)
- the basic constraint for George
- the basic constraint for the backstroke
- the constraint that requires Tyler to compete in the backstroke if Kevin completes in the freestyle
- the constraint that John must be one of the competing swimmers
- the constraint that if Ahmed competes in the butterfly, Gustov cannot compete in the breaststroke
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