Practice 1 Solutions

Fill in the blank with a word or phrase.

All probabilities will be a number ranging from zero to one (0% to 100%) .
It is recommended that an experiment be performed at least 100 times so that the theoretical and experimental probabilities get closer to one another.

For problems 3–6, use the spinner.

P(A)

Total: 6, A: 1

$P (A) = \frac{1}{6}$

P(3)

Total: 6, three: 1

$P (3) = \frac{1}{6}$

The arrow was spun twice. Determine the probability of stopping on a sum of seven.

$sum of 7 : 2 and 5, 5 and 2, 1 and 6, 6 and 1, 4 and 3, 3 and 4 = 6 outcomes total outcomes : 6 \cdot 6 = 36 P (sum of 7) = \frac{6}{36}$

$P (sum of 7) = \frac{1}{6}$

Determine the probability of the arrow stopping not on B or D.

$P (B or D) = \frac{2}{6} = \frac{1}{3} P (not B or D) = 1 - \frac{1}{3}$

$P (not B or D) = \frac{2}{3}$

Note

Another name for “not” is a complement. You can also find the answer by totaling the outcomes that are not B or D.

For problems 7–8, use the following scenario.

Jennie wrote each letter of her name on a piece of paper and placed them in a bag.

Without looking, Jennie’s friend Maria selected one piece from the bag. What is the probability that Maria will take out an ‘e’ or ‘n’?

$e or n : e, e, n, n Total outcomes : 6 P (e or n) = \frac{4}{6}$

$P (e or n) = \frac{2}{3}$

What is the probability that Maria will take out a consonant?

Consonants: j, n, n

$P (consonsant) = \frac{1}{2}$

Note

The vowels are {A, E, I, O, U}. The complement to the vowels are the consonants.

Two crayons are drawn from a box containing four blue and six red crayons. What is the probability that both will be blue?

$n P r (10, 2) = 90 Must be blue : 3 \cdot 4 = 12 P (blue, blue) = \frac{12}{90}$

$P (blue, blue) = \frac{2}{15}$

A bag contains cards labeled with the numbers 3, 4, 5, 6, and 7. One at a time, one card is randomly selected from the bag. What is the probability that the cards drawn in order will represent a number that is greater than 7000?

$n P r (5, 5) = 120$

The first number must be 7. The remaining order does not matter.

$1 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 24 P (# > 7000) = \frac{24}{120}$

$P (# > 7000) = \frac{1}{5}$

For problems 11–15, use the frequency chart.

Ali had a bag of 30 beads, six of each color: {red, white, blue, yellow, black}. The frequency chart shows her results after drawing a bead and then replacing it.

Bead	Tally
Red	24
White	19
Blue	20
Yellow	20
Black	17

Find experimental probabilities for each bead color.

Total: $24 + 19 + 20 + 20 + 17 = 100$

Bead	Experimental Probability
Red, R	$P (R) = \frac{24}{100} = 0.24$
White, W	$P (W) = \frac{19}{100} = 0.19$
Blue	$P (blue) = \frac{20}{100} = 0.20$
Yellow, Y	$P (Y) = \frac{20}{100} = 0.20$
Black	$P (black) = \frac{17}{100} = 0.17$

$P (R) = 0.24, P (W) = 0.19, P (blue) = 0.20, P (Y) = 0.20, P (black) = 0.17$

What is the most likely bead (event) to be selected according to the experiment? Explain.

$P (R) = \frac{24}{100} = 0.24$

The red bead has the highest experimental probability (0.24, or 24%) of being drawn, making it the most likely to be selected.

What is the least likely bead (event) to be selected according to the experiment? Explain.

$P (black) = \frac{17}{100} = 0.17$

The black bead has a 17% experimental probability, making it the least likely to be selected.

Determine the theoretical probability of each bead as a decimal.

Bead	Theoretical Probability
Red, R	$P (R) = \frac{6}{30} = \frac{1}{5} = 0.2$
White, W	$P (W) = \frac{6}{30} = \frac{1}{5} = 0.2$
Blue	$P (blue) = \frac{6}{30} = \frac{1}{5} = 0.2$
Yellow, Y	$P (Y) = \frac{6}{30} = \frac{1}{5} = 0.2$
Black	$P (black) = \frac{6}{30} = \frac{1}{5} = 0.2$

$P (R) = P (W) = P (blue) = P (Y) = P (black) = 0.2$
or
All the probabilities are equal.

Explain if the beads were drawn fairly by comparing the experimental and theoretical probabilities.

Sample: All of the theoretical probabilities are equal to 0.2, and all of the experimental probabilities are between 0.17 and 0.24. Because the experimental results are close to the theoretical probability, the beads were drawn fairly.

Kylin has a drawer containing five pairs of yellow socks, two pairs of blue socks, and three pairs of red socks. Explain which color of socks Kylin should expect when one pair is randomly selected.

$Total : 5 + 2 + 3 = 10 P (Y) = 0.5 = 50 % P (B) = 0.2 = 20 % P (R) = 0.3 = 30 %$

Sample: Kylin will most likely pull out a pair of yellow socks. The probability of yellow is 50%, while the probability of blue is 20%, and the probability of red is 30%.

Note

Because there are ten choices, you can write the probability as a percent without writing the fraction.

West’s shelf has a selection of books: three are by Tolkien, one by C.S. Lewis, and the two remaining are by Jane Austen. If West chooses a book at random, a book by which author is least likely to be picked? Explain.

$Total : 3 + 1 + 2 = 6 P (Tolkien) = \frac{3}{6} P (C . S . Lewis) = \frac{1}{6} P (Jane Austen) = \frac{2}{6}$

Sample: The book by C.S. Lewis is least likely since the probability of choosing that author is $\frac{1}{6} .$

For problems 18–20, state if the event is likely, unlikely, will always happen, or will never happen. Explain.

Wren flips a fair coin three times. What is the likelihood of the flips resulting in heads every time?

$Event : (H, H, H) = 1 Total outcomes : 2 \cdot 2 \cdot 2 = 8 P (H, H, H) = \frac{1}{8}$

Unlikely

The probability of flipping heads three times in a row is $\frac{1}{8}$ .

Note

When an event has a probability less than 50% the event is considered unlikely.

Shae is flipping a fair coin. What is the likelihood of getting heads or tails on the first flip?

$P (heads or tails) = 1$

Always

The coin has only two sides, so the toss will be either heads or tails.

Raphael is rolling a 9-sided die numbered 1 through 9. What is the likelihood of rolling a two-digit number?

$P (two - digit number) = 0$

Never

There are no two-digit numbers on the die.