Question

learning goal from lesson 20.3
i can calculate the probability of compound events and interpret the
solution in context.

1. a bag contains tiles with the letters shown at the right. 3 tiles are

drawn at random and are not replaced. (1 point total)
a) what is the probability that the order of the tiles are b, then a, and then a again? choose the option that
demonstrates the correct work to find the probability
□ a. $\frac { 2 } { 12 } \cdot \frac { 3 } { 11 } \cdot \frac { 3 } { 10 }$
□ b. $\frac { 2 } { 12 } \cdot \frac { 3 } { 11 } \cdot \frac { 2 } { 10 }$
□ c. $\frac { 2 } { 12 } \cdot \frac { 3 } { 12 } \cdot \frac { 2 } { 12 }$
b) based on the work below, what probability is trying to be found?
$\frac { 3 } { 12 } \cdot \frac { 2 } { 11 } \cdot \frac { 1 } { 10 }$
□ a. the probability that the first 3 tiles are different.
□ b. the probability that the first 3 tiles are in the order a, b, c.
□ c. the probability that the first 3 tiles are all as.

Explanation:

a) Step1: Calculate the probability of drawing B first

There are 12 tiles in total and 2 B - tiles. The probability of drawing a B first is \(P(B)=\frac{2}{12}\).

a) Step2: Calculate the probability of drawing A second

After drawing a B, there are 11 tiles left. There are 3 A - tiles. So the probability of drawing an A second is \(P(A)=\frac{3}{11}\).

a) Step3: Calculate the probability of drawing A third

After drawing B and A, there are 10 tiles left. There are 2 A - tiles left. So the probability of drawing an A third is \(P(A)=\frac{2}{10}\).

a) Step4: Use the multiplication rule for dependent events

The probability of the compound event (B then A then A) is \(P = \frac{2}{12}\times\frac{3}{11}\times\frac{2}{10}\).

b) Step1: Analyze the formula \(\frac{3}{12}\cdot\frac{2}{11}\cdot\frac{1}{10}\)

The first factor \(\frac{3}{12}\) is the probability of drawing one of a kind (say \(x\)) from 12 tiles. The second factor \(\frac{2}{11}\) is the probability of drawing a different tile from the remaining 11 tiles. The third factor \(\frac{1}{10}\) is the probability of drawing a tile different from the first two from the remaining 10 tiles.

Answer:

a) B. \(\frac{2}{12}\cdot\frac{3}{11}\cdot\frac{2}{10}\)
b) A. The probability that the first 3 tiles are different.