**High School Number Sense Lesson 68: Probability with Coins**

We will now enter the wonderful world of

**probabilities**(not to be confused with Odds, which we will cover in a later lesson). The most relatable item dealing with probabilities is a coin, which is designed to give someone a 50% probability of flipping a heads, and a 50% probability of flipping a tails. On number sense tests, you will be asked the probability of getting a specific result when flipping multiple coins. This concept appeared

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

**question # 64**.

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These problems use the concept of

**Combination**, with some extra calculation. As an overview, the probability of an event happening is the number of favorable outcomes divided by the total number of outcomes. This can be represented more visually:

**# favorable outcomes**

P(E) = ------------------------------

# total outcomes

P(E) = ------------------------------

# total outcomes

__: On number sense tests, probabilities are usually expressed as fractions, but they could also be represented as decimals or percents. Pay close attention to what each test question requires.__

**Note**When flipping multiple coins (or flipping one coin multiple times), the number of favorable outcomes is going to be a combination:

_{n}C_{r}**n**is the number of coins (or the number of coin flips), and**r**is the number of favorable results

So, to calculate a probability of specific outcomes of a coin toss, follow these steps:

- For the numerator, take the
**combination**of the # of coin tosses, taken the # of favorable results at a time. - For the denominator, take
**2 to the nth power**, where n is the number of coin tosses. - Reduce the fraction if possible.

**Example 1: The probability of flipping 3 coins and getting all tails is ___**

_{3}C

_{3}= 3!/(3!0!) = 6/(6 x 1) = 6/6 =

**1**.

2. Denominator: 2

^{3}=

**8**.

3. The answer is

**1/8**.

**Example 2: The probability of flipping 4 coins and getting all heads is ___**

_{4}C

_{4}= 4!/(4!0!) = 1/0! = 1/1 =

**1**.

2. Denominator: 2

^{4}=

**16**.

3. The answer is

**1/16**.

**Example 3: When flipping 4 coins, what is the probability of getting 3 tails and 1 head? ___**

_{4}C

_{3}= 4!/(3!1!) = (4 x 3!)/3!1! = 4/1! =

**4**.

2. Denominator: 2

^{4}=

**16**.

3. The answer is 4/16 =

**1/4**.

**Example 4: Five coins are tossed. What is the probability of getting two heads and three tails? ___**

_{5}C

_{3}= 5!/(3!2!) = (5 x 4 x 3!)/3!2! = (5 x 4)/2! = 20/2 =

**10**.

2. Denominator: 2

^{5}=

**32**.

3. The answer is 10/32 =

**5/16**.

**Example 5: Six coins are tossed. What is the probability of getting 4 heads and 2 tails? ___**

_{6}C

_{4}= 6!/(4!2!) = (6 x 5 x 4!)/4!2! = (6 x 5)/2! = 30/2 =

**15**.

2. Denominator: 2

^{6}=

**64**.

3. The answer is

**15/64**.

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

**Up Next for High School: Odds**