Question

A cereal company is putting 1 of 4 prizes in each box of cereal. The prizes are evenly distributed so the

probability of winning any given prize is always 1/4.

Lily wonders how many boxes she should expect to buy to get all 4 prizes. She carried out 20 trials of a

simulation and her results are shown below. Each dot represents how many boxes it took to get all 4 prizes in

that trial.

4

5

6

7

8

9

10 11 12 13 14 15

# of boxes purchased

Use her results to estimate the probability that it takes fewer than 7 boxes to get all 4 prizes.

Give your answer as either a fraction or a decimal.

P(fewer than 7 boxes) -

Answer 1

To calculate the probability of needing fewer than 7 boxes to collect all 4 prizes, tally up trials with fewer than 7 boxes and divide by the total number of trials. In this case, it is 3 out of 20 trials, resulting in a probability of 0.15.

Step-by-step explanation:

The student has simulated to find out how many boxes of cereal she needed to buy to collect all 4 prizes, and from the 20 trials, she observed the number of boxes purchased in each trial. To estimate the probability that it takes fewer than 7 boxes to get all 4 prizes, we count the number of trials in which the number of boxes purchased is less than 7 and divide it by the total number of trials (which is 20).

The results from the simulation trials show 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. From these results, we only look at the outcomes of 4, 5, and 6 to determine our answer since we are interested in occurrences of fewer than 7 boxes.

Let's count the dots (trials) that resulted in purchasing fewer than 7 boxes: there is one at 4, one at 5, and one at 6. This gives us 3 trials out of 20 where the number of boxes purchased was less than 7.

Therefore, the probability that it takes fewer than 7 boxes to get all 4 prizes is 3/20 or 0.15.

Answer 2

`P(x < 7) = 0.383`

Step-by-step explanation:

Given

See attachment for plot

Required

Determine the probability that few than 7 boxes will be needed.

This is represented as:

`P(x < 7)`

And the solution is:

`P(x < 7) = P(x = 4) + P(x = 5) + P(x = 6)`

Each probability is calculated by dividing the number of dots by the total dots.

So, we have:

`P(x < 7) = (n(x = 4) + n(x = 5) + n(x = 6))/(Total)`

From the attached, the formula becomes:

`P(x < 7) = (4 + 6 + 3)/(4+6+3+4+4+6+3+2+1+2+0+1)`

`P(x < 7) = (13)/(36)`

`P(x < 7) = 0.361`