INTERMEDIATE PROBABILITY PROBLEMS

Concept: Probability, chance, and odds are different ways of expressing the likelihood that an event will occur.

Probability is the likelihood that a particular event will occur. Probability is written as a ratio. Probability is the ratio of the number of favorable outcomes to the number of possible outcomes

Chance is a way of expressing the likelihood of an event. It is the probability of an event expressed as a percent. We can use the word chance and a percent to describe a probability. If a probability is expressed as ¼, we can say that the chance of that thing happening is 25 percent.

Percent chance of a probability of ¼ = 1 4 = 0.25 = 25%

Odds is the ratio of the number of favorable outcomes to the number of unfavorable outcomes.

Note that odds are usually expressed by using the word “to”, or when writing the expression mathematically by using a colon, not by using a division line. Odds are reduced just as are other ratios.

odds = number of favorable outcomes TO number of unfavorable outcomes.

or written mathematically

X : Y

( When X = number of favorable outcomes, and Y = number of unfavorable outcomes )

Unfavorable outcomes can be calculated by subtracting the favorable outcomes from the possible outcomes.

Calculation for Unfavorable Outcomes

possible outcomes - favorable outcomes. unfavorable outcomes

Steps for Solving Intermediate Probability Problems

Step 1 | Determine what kind of question you are being asked. Are you being asked to find a probability fraction, a chance (percent probability), or the odds?
Step 2 | If you are being asked to find a probability fraction, set up the probability fraction. Reduce as necessary.
Step 3 | If you are being asked to find a percent probability, set up the probability fraction as defined in Step 2. Reduce as necessary. Solve the division. Convert the decimal answer to a percent.
Step 4 | If you are being asked to find the odds, first set up a probability fraction. Reduce as necessary. Set up the ratio of favorable outcomes to unfavorable outcomes.
Step 5 | Write your final answer.

Example 1: What is the probability fraction of the spinner pointing two times in a row to a 49?

Step 1 | Determine what kind of question you are being asked. Are you being asked to find a probability fraction, a chance (percent probability), or the odds? | probability fraction
Step 2 | If you are being asked to find a probability fraction, set up the probability fraction. Reduce as necessary. |
| There are seven wedges in the circle, but one of the wedges is twice the size of all the others. That wedge must be counted as two wedges, so for the sake of this problem we will say that there are eight wedges. There is only one wedge marked 49. Therefore, the number of favorable outcomes for ONE spin is 1 and the number of possible outcomes for ONE spin is 8. | 1 8
| The question, however, is “What is the probability fraction of the spinner pointing two times in a row to a 49?” To find the solution we do not simply multiply 1/8 by 2. The possibilities are much larger. Multiply the probability fraction by itself. |
Step 3 | If you are being asked to find a percent probability, set up the probability fraction as defined in Step 2. Reduce as necessary. Solve the division. Convert the decimal answer to a percent. | Does not apply.
Step 4 | If you are being asked to find the odds, first set up a probability fraction. Reduce as necessary. Set up the ratio of favorable outcomes to unfavorable outcomes. | Does not apply.
Step 5 | Write your final answer. |

Example 2: What is the percent probability of drawing a Diamond face card 2 times?

Step 1 | Determine what kind of question you are being asked. Are you being asked to find a probability fraction, a chance (percent probability), or the odds? | chance (percent probability)
Step 2 | If you are being asked to find a probability fraction, set up the probability fraction. Reduce as necessary. | Does not apply.
Step 3 | If you are being asked to find a percent probability, set up the probability fraction as defined in Step 2. |
| Let’s first solve for drawing a Diamond face card one time. There are three diamond face cards in a deck of cards, so the number of favorable outcomes is 3. There are 52 cards in a deck of cards, so the number of possible outcomes is 52. The probability fraction for drawing a Diamond face card once is 3 out of 52. |
| The question, however, is “What is the percent probability of drawing a Diamond face card 2 times?” To find the solution we do not simply multiply 3/52 by 2. Since we withdrew 1 card, the number of favorable outcomes is now only 2. And again, since we withdrew 1 card, the number of possible outcomes is now 51. |
| Cross cancel before multiplying. |
| Reduce as necessary. | Not necessary.
| Solve the division. |
| Convert the decimal answer to a percent. | 0.0023 = 0.23%
Step 4 | If you are being asked to find the odds, first set up a probability fraction. Reduce as necessary. Set up the ratio of favorable outcomes to unfavorable outcomes. | Does not apply.
Step 5 | Write your final answer. | 0.23%

Example 3: The die was rolled 5 times. What are the odds of rolling a number less than or equal to a 3 on every roll?

Step 1 | Determine what kind of question you are being asked. Are you being asked to find a probability fraction, a chance (percent probability), or the odds? | odds
Step 2 | If you are being asked to find a probability fraction, set up the probability fraction. Reduce as necessary. | Does not apply.
Step 3 | If you are being asked to find a percent probability, set up the probability fraction as defined in Step 2. Reduce as necessary. Solve the division. Convert the decimal answer to a percent. | Does not apply.
Step 4 | If you are being asked to find the odds, first set up a probability fraction. |
| Let’s first determine the probability fraction for one roll of the die. The number of favorable outcomes is 3. The number of possible outcomes is 6. | 3 6
| Reduce the fraction. | 1 2
| Since the die is rolled 5 times, we take the fraction to the fifth power. | (½)5
| Reduce. Reduce as necessary. | ½ x ½ x ½ x ½ x ½ =1/32
| Set up the ratio of favorable outcomes to unfavorable outcomes. | 1 : (32 – 1) 1 : 31
Step 5 | Write your final answer. | 1 : 31