3) the mega millions® is a multi - state lottery game. there are drawings twice a week for the “jackpot”. players may pick six numbers from two separate pools of numbers - five different numbers from 1 to 70 (the white balls) and one number from 1 to 25 (the gold mega ball). you win the jackpot by matching all six winning numbers in a drawing. you can win $1 million dollars by matching all 5 of the white balls together. you compile a data set of all the past white ball picks (20 drawings, 5 white balls drawn each time is 100 numbers total). a histogram of the data is below. use the histogram to answer the questions that follow.
a) the lottery numbers can be any number between 1 and 70. why do the intervals start at 0 and end at 80?
b) how many lottery numbers between 1 and 29 were drawn?
c) how many times did the number 70 get drawn? how do you know?
d) if the lottery is fair, draw a histogram you would expect to see: (a rough sketch is all that is needed)

Question

Accepted Answer

### Part (a)
# Explanation:
## Step1: Analyze number range
Lottery numbers are 1 - 70. Intervals start at 0 (to cover numbers just below 1, though none exist here) and end at 80 (to cover numbers up to 70 with a buffer, ensuring all 1 - 70 are within the interval range [0,80), and the histogram bins are properly defined without cutting off the upper limit of 70.
# Answer:
To ensure the interval range ($0 - 80$) fully encompasses the lottery number range ($1 - 70$), so all relevant numbers are included in the histogram bins without truncation.

### Part (b)
# Explanation:
## Step1: Identify intervals for 1 - 29
The intervals for 1 - 29 would be $0 - 10$, $10 - 20$, $20 - 30$ (since 29 is within $20 - 30$).
## Step2: Read frequencies from histogram
- For $0 - 10$: Frequency ≈ 17 (from the first bar, height ~17)
- For $10 - 20$: Frequency ≈ 15 (second bar, height ~15)
- For $20 - 30$: Frequency ≈ 14 (third bar, height ~14)
## Step3: Sum the frequencies
Sum = $17 + 15 + 14 = 46$
# Answer:
46

### Part (c)
# Explanation:
## Step1: Identify the interval for 70
The number 70 is in the interval $70 - 80$.
## Step2: Read frequency of $70 - 80$
From the histogram, the bar for $70 - 80$ has a frequency of approximately 5 (height ~5). Since 70 is the lower bound of this interval, we assume the count for 70 is part of this interval's frequency (assuming uniform distribution within the interval, or the bar represents the count for numbers 70 - 80, so 70's count is included in this 5).
# Answer:
Approximately 5 times. We know this by looking at the frequency of the $70 - 80$ interval in the histogram, as 70 is within this interval.

### Part (d)
# Explanation:
## Step1: Understand "fair" lottery
A fair lottery means each number (1 - 70) has an equal chance of being drawn. So the frequency of each number should be approximately equal.
## Step2: Calculate expected frequency per interval
Total numbers drawn: 100 (20 drawings × 5 balls). Number of intervals: Let's assume intervals are $0 - 10$, $10 - 20$,..., $70 - 80$ (8 intervals). But since numbers are 1 - 70, we can consider 7 intervals of 10 numbers (1 - 10, 11 - 20,..., 61 - 70) and one interval 71 - 80 (with 0 frequency). For 1 - 70 (70 numbers), expected frequency per 10 - number interval: $\frac{100}{70}\times10\approx14.29$ (or per number: $\frac{100}{70}\approx1.43$ per number, so per 10 - number interval ~14 - 15).
## Step3: Sketch the histogram
Draw a histogram with intervals $0 - 10$, $10 - 20$,..., $70 - 80$. The bars for $0 - 10$ (but numbers 1 - 10), $10 - 20$ (11 - 20),..., $60 - 70$ (61 - 70) should have approximately equal height (around 14 - 15), and the bar for $70 - 80$ (71 - 80) should have height 0 (since no numbers 71 - 80 are drawn). The bars should be relatively flat (uniform distribution) across the 1 - 70 range.
# Answer:
A rough sketch would have 8 intervals ($0 - 10$ to $70 - 80$). The first 7 intervals (covering 1 - 70) have approximately equal height (since each number is equally likely, so each 10 - number block has similar frequency), and the last interval ($70 - 80$) has height 0 (as no numbers 71 - 80 are in the lottery). The bars for $0 - 10$ to $60 - 70$ are flat (uniform) with height around 14 - 15 (since $100$ total draws, $70$ numbers, so ~1.43 per number, ~14.3 per 10 - number interval).

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QUESTION IMAGE

Question

Explanation:

Part (a)

Step1: Analyze number range

Step1: Identify intervals for 1 - 29

Step2: Read frequencies from histogram

Step3: Sum the frequencies

Step1: Identify the interval for 70

Step2: Read frequency of \(70 - 80\)

Answer:

Part (b)