UpStudy Free Solution:

To determine the expected number of winners out of 100 players based on the data given, we first need to calculate the overall winning rate from the three games.

Step 1: Calculate the total number of players and total number of winners

\[\text { Total number of players} = 52 + 77 + 61 = 190\]

\[\text { Total number of winners} = 14 + 25 + 18 = 57\]

Step 2: Calculate the overall winning rate

\[\text { Winning rate} = \frac { \text { Total number of winners} } { \text { Total number of players} } = \frac { 57} { 190} \]

Step 3: Calculate the expected number of winners for 100 players

\[\text { Expected number of winners} = \text { Winning rate} \times 100 = \left ( \frac { 57} { 190} \right ) \times 100\]

\[\text { Expected number of winners} \approx 0.3 \times 100 = 30\]

Therefore, based on the data recorded in the table, you would expect approximately 30 winners if there are 100 players.

Answer: 30

Supplemental Knowledge

Understanding how to calculate expected outcomes based on given probabilities is a fundamental concept in statistics. Here’s a deeper insight into these concepts:

1. Probability Basics:

- Probability is a measure of the likelihood that an event will occur. It ranges from 0 (impossible event) to 1 (certain event).

- The probability \(P\) of an event occurring is calculated as:

\[P = \frac { \text { Number of favorable outcomes} } { \text { Total number of possible outcomes} } \]

2. Expected Value:

- The expected value (EV) is a key concept in probability and statistics, representing the average outcome if an experiment is repeated many times.

- For discrete random variables, the expected value \(E( X) \) can be calculated by summing the products of each outcome and its corresponding probability:

\[E( X) = \sum ( x_ i \times P( x_ i) ) \]

3. Example Calculation:

- Given data:

- Total number of players across three games = \(52 + 77 + 61 = 190\)

- Total number of winners across three games = \(14 + 25 + 18 = 57\)

- Calculate the overall winning rate:

\[\text { Winning rate} = \frac { 57} { 190} \approx 0.3\]

- To find the expected number of winners out of 100 players:

\[E( \text { Winners} ) = ( \text { Winning rate} ) \times ( \text { Number of players} ) \\ = 0.3 \times 100\\ = 30\]

Thus, based on this winning rate, you would expect approximately \(30\) winners out of \(100\) players.

Acknowledging statistical concepts like expected value can dramatically hone your analytical abilities! UpStudy offers comprehensive platform that simplifies complex topics while offering step-by-step solutions across various subjects; therefore it makes an ideal learning companion.