The probability that you will win a game is 
.

If you play the game 783 times, what is the most likely number of wins?
(Round answer to one decimal place.)
 = 


Let 
 represent the number of games (out of 783) that you win. Find the standard deviation for the probability distribution of 
.
(Round answer to two decimal places.)
σ = 


Use the range rule of thumb to find the minimum usual value 
 and the maximum usual value 
.
(Round answers to one decimal place.)
Minimum Usual Value = 

Maximum Usual Value = 


To solve this problem, we need to go through three steps: finding the most likely number of wins, calculating the standard deviation for the probability distribution, and using the range rule of thumb to find the minimum and maximum usual values.

### 1. **Finding the Most Likely Number of Wins**
The number of wins follows a binomial distribution, where:
- \( n = 783 \) (number of games),
- \( p \) is the probability of winning a single game.

The **most likely number of wins** is given by the expected value \( E(X) \), which can be calculated as:
\[
E(X) = n \cdot p
\]

### 2. **Finding the Standard Deviation (σ)**
The standard deviation for a binomial distribution is calculated as:
\[
\sigma = \sqrt{n \cdot p \cdot (1 - p)}
\]
where \( n = 783 \) and \( p \) is the probability of winning a single game.

### 3. **Using the Range Rule of Thumb**
The range rule of thumb states that the usual values in a probability distribution fall within:
\[
\text{Minimum Usual Value} = E(X) - 2\sigma
\]
\[
\text{Maximum Usual Value} = E(X) + 2\sigma
\]

Now, please provide the value of \( p \) (the probability of winning a single game), and I can help you calculate the answers.

The probability that you will win a game is . If you play the game 783 times, what is the most likely number of wins? (Round answer to one decimal place.) Let X represent the number of games (out of 783) that you win. Find the standard deviation for the probability distribution of X. (Round answer to two decimal places.) Use the range rule of thumb to find the minimum usual value and the maximum usual value. (Round answers to one decimal place.)

Calculate the most likely number of wins for 783 games, find the standard deviation, and use the range rule of thumb to find the minimum and maximum usual values. This problem covers key probability concepts such as binomial distribution, expected value, and standard deviation. Suitable for students in Grades 10-12.

Math Problem Statement

If you play the game 783 times, what is the most likely number of wins? (Round answer to one decimal place.)

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