**High School Number Sense Lesson 69: Odds**

On Saturday we introduced probabilities with our

**ProbCoin**lesson. Today we will tweak it a little and discuss Odds. This concept appeared

**4 times**last year, with a median placement at

**question # 65**.

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The odds of some event happening are different from the probability of it happening. You'll remember that the probability is equal to the number of

**favorable outcomes**, divided by the number of

**total outcomes**. The odds are equal to the number of

**favorable outcomes**, divided by the number of

**unfavorable outcomes**. This can be represented more visually:

**# favorable outcomes**

Odds = -----------------------------------

# unfavorable outcomes

Odds = -----------------------------------

# unfavorable outcomes

**How to Calculate Odds:**

- For the numerator, just like with probabilities, take the
**combination**, n taken r at a time. - For the denominator, take
**2 to the nth power**, and**subtract the result in Step 1**. - Reduce the fraction if possible.

**Example 1: The odds of randomly choosing a vowel from the letters of the word ROMAN is ___**

- Numerator: there are
**2**vowels in the word. - Denominator: there are
**3**non-vowels in the word. - The answer is
**2/3**, which does not reduce.

**Example 2: If an urn contains 3 red marbles, 4 blue marbles and 8 orange marbles, then the odds of choosing a red marble is ___**

- Numerator: there are
**3**red marbles. - Denominator: there are 4 + 8 =
**12**non-red marbles. - The answer is
**3/12**, which reduces to**1/4**.

**Example 3: The probability of winning is 40%. The odds of winning is ___%.**

- Change 40% to a fraction: 2/5. The numerator is
**2**. - Denominator: 5 - 2 =
**3**. - The answer is
**2/3**, which does not reduce.

**Example 4: The odds of losing are 4 to 7. The probability of winning is ___**

- The probability of losing is 4/(4 + 7) =
**4/11**. - The probability of winning is (1 - 4/11) =
**7/11**. - Think of this question as saying that there are 4 ways of losing to every 7 ways of winning. Out of 11 total outcomes, 7 are winners. Thus the
**7/11**.

**Example 5: When tossing 5 coins, what are the odds of getting 1 tail and 4 heads?**

- Numerator: there are
**5**ways of getting 1 tail (THHHH, HTHHH, HHTHH, HHHTH, and HHHHT). - Denominator: 2 to the 5th power is
**32**, minus the**5**results from Step 1, =**27**. - The answer is
**5/27**, which does not reduce.

**Check back soon for a free worksheet to help you practice Odds.**

**Up Next for High School: EstPiPower**