Exercises: Approximate the probability of a chance event by collecting data - Common Core Math - Grade 7

A

Recall / Warm-Up

1.

A spinner is spun 50 times. It lands on 'blue' 18 times. Which fraction correctly represents the relative frequency of blue?

A.

$\frac{18}{32}$

B.

$\frac{18}{50}$

C.

$\frac{32}{50}$

D.

$\frac{50}{18}$

2.

Express $\frac{34}{80}$ as a decimal. Round to the nearest thousandth.

3.

A store sells 300 items in a day. If 0.6 of the items are clothing, how many clothing items were sold?

B

Fluency

1.

A coin is flipped 80 times. Heads appeared 34 times. Compute the experimental probability of heads as a decimal.

2.

A standard 6-sided die is rolled 180 times. How many times would you expect to roll a 4?

3.

Which statement correctly defines experimental probability?

A.

The theoretical fraction of outcomes that should produce the event based on the structure of the experiment.

B.

The count of favorable outcomes in an experiment.

C.

The fraction of trials in which the event occurred: (number of occurrences) $\div$ (total number of trials).

D.

The probability assigned to an event before any trials are conducted.

4.

A spinner is spun 120 times. Results: Red = 42 times, Blue = 38 times, Green = 40 times. Compute the experimental probability of Red as a fraction in lowest terms.

5.

Which statement best describes what 'long-run relative frequency' means in probability?

A.

After running an experiment many times, the experimental probability will eventually equal the theoretical probability exactly.

B.

As the number of trials increases, the experimental probability gets closer and closer to the theoretical probability.

C.

Long-run relative frequency only applies to coin flips, not to other probability experiments.

D.

The relative frequency stays constant after the first 50 trials.

C

Varied Practice

D

Word Problems

1.

A student draws colored tiles from a bag with replacement 50 times. Results: Blue = 24 times, Red = 16 times, Yellow = 10 times.

1.

Compute the experimental probability of drawing a Blue tile as a decimal.

2.

The bag was designed so that the theoretical probability of blue is 0.40. Compare this to the experimental result of 0.48. Is this discrepancy concerning? Explain your reasoning.

2.

The theoretical probability of rolling an even number on a standard 6-sided die is $\frac{1}{2}$ . If the die is rolled 240 times, predict the approximate number of even-number rolls.

3.

A student flips a fair coin 10 times and gets tails 8 times. The student says: 'The next few flips must be mostly heads to balance things out.' (a) What is the Gambler's Fallacy? (b) Why is the student's reasoning incorrect?

4.

A student rolls a standard 6-sided die 6 times and does not roll a 6 in any of the 6 rolls. The student concludes: 'This die is definitely unfair — the 6 is blocked.' Is this conclusion justified?

A.

Yes — rolling zero 6s in 6 tries proves the die is biased.

B.

No — 6 rolls is too small a sample to conclude anything. Rolling zero 6s in 6 tries happens about 1 in 3 times even with a fair die. Hundreds of rolls would be needed to detect a true bias.

C.

Yes — after 6 rolls, the student should have rolled exactly one 6 based on probability.

D.

No — the student should have rolled the die at least 36 times before drawing any conclusion.

E

Error Analysis

1.

A student knows that P(red tile) = 0.25. The student draws tiles 40 times with replacement and gets 7 red tiles. The expected count was $0.25 \times 40 = 10$ reds.

The student writes: "This experiment must be wrong — I should have gotten exactly 10 reds since $0.25 \times 40 = 10$ . The data is invalid."

What error did the student make? What does P × n actually tell us?

A.

The student computed the expected value incorrectly: $0.25 \times 40 \ne 10$ .

B.

The expected frequency P × n is a prediction, not a guarantee. Getting 7 instead of 10 in 40 trials is normal random variation — the experiment is valid.

C.

The student should have run 40 trials 4 times and averaged the results.

D.

The experiment is invalid because 7 ≠ 10.

2.

A student is asked to estimate the probability of a spinner landing on green. After 90 spins, the spinner landed on green 27 times.

The student writes: "P(green) is about 27."

What error did the student make? What is the correct experimental probability?

A.

The student should have divided the total spins by the green count: $90 \div 27 \approx 3.33$ .

B.

The student reported the count (27) instead of the proportion. P(green) $= \frac{27}{90} = \frac{3}{10} = 0.30$ .

C.

The student should have compared green to the other outcomes first.

D.

There is no error — 27 is an acceptable expression of experimental probability.

F

Challenge

1.

A die is rolled 600 times. The results are:
Face 1: 89 times, Face 2: 95 times, Face 3: 102 times,
Face 4: 105 times, Face 5: 98 times, Face 6: 111 times.

(a) Compute the experimental probability for each face. Round to three decimal places.
(b) Verify that the six probabilities sum to 1.
(c) The theoretical probability of each face is $\frac{1}{6} \approx 0.167$ . Compare each experimental value to 0.167.
(d) Based on this data, does the die appear to be fair or biased? Explain your reasoning.

2.

You want to estimate the probability that a thumbtack lands point-up when dropped.