6. Multiplying Probabilities
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| Class | Freshman | Sophomore | Junior | Senior | Total |
|---|---|---|---|---|---|
| Attended | 48 | 90 | 224 | 254 | 616 |
| Not attended | 182 | 141 | 36 | 8 | 367 |
| Total | 230 | 231 | 260 | 262 | 983 |
From the bottom right entry we can see, there were 983 students in North Coast High School. If we divide the numbers with this total, we get the probabilities associated with the cells.
Notice that 983 is a prime number, so the fractions cannot be reduced.
| Class | Freshman | Sophomore | Junior | Senior | Total |
|---|---|---|---|---|---|
| Attended | 48/983 | 90/983 | 224/983 | 254/983 | 616/983 |
| Not attended | 182/983 | 141/983 | 36/983 | 8/983 | 367/983 |
| Total | 230/983 | 231/983 | 260/983 | 262/983 | 1 |
We are interested in the probability that a randomly selected student has not attended a game, given that he or she is a freshman. Let's use the Conditional Probability Formula to write this probability as the quotient of two unconditional probabilities. P(not attended | freshman)=P(not attended and freshman)/P(freshman) Let's use the color we used in the formula to highlight the relevant entries of the table.
| Class | Freshman | Sophomore | Junior | Senior | Total |
|---|---|---|---|---|---|
| Attended | 48/983 | 90/983 | 224/983 | 254/983 | 616/983 |
| Not attended | 182/983 | 141/983 | 36/983 | 8/983 | 367/983 |
| Total | 230/983 | 231/983 | 260/983 | 262/983 | 1 |
If we substitute these values in the formula, we get the answer to the question. P(not attended | freshman)=182/983/230/983=91/115≈ 0.791 The probability that a randomly selected student has not attended a football game, given that he or she is a freshman, is 91115, or about 0.791.
P(upperclassman | attended)=P(uperclassman and attended)/P(attended)
Let's use the color we used in the formula to highlight the relevant entries of the table.
Note that both juniors and seniors are upperclassman, so we need to consider both cells when we calculate the probability in the numerator.
| Class | Freshman | Sophomore | Junior | Senior | Total |
|---|---|---|---|---|---|
| Attended | 48/983 | 90/983 | 224/983 | 254/983 | 616/983 |
| Not attended | 182/983 | 141/983 | 36/983 | 8/983 | 367/983 |
| Total | 230/983 | 231/983 | 260/983 | 262/983 | 1 |
Since there is no student who is both a junior and a senior, the probability of a student being an upperclassman who has attended a game is the sum of the corresponding junior and senior probabilities. 224/983+ 254/983=224+254/983= 478/983 If we substitute the values in the conditional Probability Formula, we get the answer to the question. P(upperclassman | attended)=478/983/616/983=239/308≈ 0.776 The probability that a randomly selected student is an upperclassman, given that he or she has attended a football game, is 239308, or about 0.776.