An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Calculate the total number of series: N = 20 + 23 + 25 + 31 = 99 .
Calculate the probability for each value of x: P ( x ) = N f re q u e n cy ( x ) .
Compute the mean: E ( X ) = 4 ∗ P ( 4 ) + 5 ∗ P ( 5 ) + 6 ∗ P ( 6 ) + 7 ∗ P ( 7 ) = 99 562 ≈ 5.7 .
The mean number of games played is 5.7 .
Explanation
Understand the problem We are given a frequency table representing the number of games played in each series of an annual tournament. Our goal is to compute the mean of the random variable X, which represents the number of games played.
Calculate the total number of series First, we need to find the total number of series. We sum the frequencies for each number of games played: N = 20 + 23 + 25 + 31 = 99 So, there are a total of 99 series.
Calculate the probabilities Next, we calculate the probability for each value of X (number of games played) by dividing its frequency by the total number of series: P ( X = 4 ) = 99 20 ≈ 0.2020 P ( X = 5 ) = 99 23 ≈ 0.2323 P ( X = 6 ) = 99 25 ≈ 0.2525 P ( X = 7 ) = 99 31 ≈ 0.3131
Compute the mean Now, we compute the mean (expected value) of the random variable X using the formula: E ( X ) = ∑ [ x ∗ P ( x )] = 4 ∗ P ( 4 ) + 5 ∗ P ( 5 ) + 6 ∗ P ( 6 ) + 7 ∗ P ( 7 ) E ( X ) = 4 ∗ 99 20 + 5 ∗ 99 23 + 6 ∗ 99 25 + 7 ∗ 99 31 E ( X ) = 99 80 + 115 + 150 + 217 = 99 562 ≈ 5.676767... Rounding to one decimal place, we get 5.7.
Interpret the mean The mean of the random variable X is approximately 5.7 games. This represents the average number of games played in a series over the period from 1923 to 2022.
Final Answer Therefore, the mean number of games played is approximately 5.7 games.
Examples
Understanding the average number of games in a tournament series can help in planning resources, scheduling, and predicting the duration of future tournaments. For instance, if a tournament organizer knows that the average series length is about 5.7 games, they can better estimate the total number of days needed for the tournament, the number of umpires required, and the potential revenue from ticket sales. This statistical measure provides valuable insights for effective tournament management and decision-making.
In 30 seconds, a device delivering 15.0 A of current allows approximately 2.81 × 10^21 electrons to flow through it. This is calculated using the relationship between current, charge, and the fundamental charge of an electron. Thus, we find that the total amount of flow of electric charge constitutes a vast number of electrons.
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