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Using Algebra in Statistics and Probability Study Guide (page 2)

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Updated on Oct 3, 2011

Finding a Range

The range of a data set is the difference between its greatest value and its least value. In the set {1, 3, 5, 5, 7}, the range is 6, because the greatest value is 7, the least value is 1, and 7 – 1 = 6.

Now we are told that a new value has been added to the set, and the range is now 10. Use x to represent the new value. What could that value be? Because the range has changed, the new value is now either the smallest value in the set or the largest value in the set. If the new value is the smallest value, then the range, 10, is equal to 7 – x. If the new value is the largest value, then the range is equal to x – 1:

7 – x = 10 x – 1 = 10
x = 3 x = 11
x = –3

The new value is either –3 or 11.

Probability

Probability is the likelihood that an event or events will occur. We usually write probability as a fraction. The denominator is the total number of possibilities, and the numerator is the number of possibilities that make an event true. For example, a coin has two sides, heads and tails, so the probability of a coin landing on heads is , because there are two possibilities (heads or tails) and only one possibility that makes the event "landing on heads" true.

Example

A gumball machine contains 12 red gumballs, 8 green gumballs, 5 yellow gumballs, and an unknown number of orange gumballs. If the probability of selecting a green gumball is , how many orange gumballs are in the machine?

The probability of selecting a green gumball is . We know that this probability is equal to the number of green gumballs, 8, divided by the total number of gumballs. We can let x represent the number of orange gumballs. The total number of gumballs is equal to 12 + 8 + 5 + x = 25 + x. The fraction is equal to , because both fractions represent the probability of selecting a green gumball. When two fractions equal each other, we can cross multiply: Multiply the numerator of the first fraction by the denominator of the second fraction, and multiply the numerator of the second fraction by the denominator of the first fraction. Then, set those products equal to each other:

      =
      25 + x = 32

Now we have an equation we can solve. By subtracting 25 from both sides, we find that x = 7. There are 7 orange gumballs in the machine.

Tip:

When a new value is added to a data set, that new value will likely change the denominator of probabilities for the data set, and you will have to add x to the denominator of these probabilities. If you are looking for the probability of selecting that new value, you may have to add x to the numerator of these probabilities, too.

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