Question

learning goal from lesson 19.3
i can calculate the probabilities of compound events using
permutations and combinations.
lesson 19.3 checkpoint
□ once you have completed the above problems and checked your solutions, complete the lesson checkpo
below.
□ complete the lesson reflection above by circling your current understanding of the learning goal.

1. a bag contains 26 tiles, each with a different letter of the alphabet written on it. you choose 3 tiles from the

bag without looking. what is the probability that you choose the tiles with the letters a, b, and c?
□ a. $\frac{1}{15600}$
□ b. $\frac{1}{2600}$
□ c. $\frac{3}{2600}$
□ d. $\frac{3}{1300}$

1. there are 12 peaches and 8 bananas in a fruit basket. you get a snack for yourself and three of your friends

by choosing four of the pieces of fruit at random. what is the probability that all 4 are peaches?

1. a school has 5 spanish teachers and 4 french teachers. the schools principal randomly chooses 2 of the

teachers to attend a conference. what is the probability that the principal chooses 2 spanish teachers?
□ a. $\frac{2}{_{9}c_{2}}=\frac{1}{18}$
□ b. $\frac{_{4}c_{2}}{_{9}c_{2}}=\frac{1}{6}$
□ c. $\frac{_{5}c_{4}}{_{9}c_{2}}=\frac{5}{36}$
□ d. $\frac{_{5}c_{2}}{_{9}c_{2}}=\frac{5}{18}$

Explanation:

Step1: Calculate the total number of ways to choose 3 tiles from 26

The formula for combinations is \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 26\) and \(r=3\).

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There is only 1 way to choose the tiles with letters \(A\), \(B\), and \(C\) (since we want that specific combination).

Step2: Calculate the probability

The probability \(P\) of an event is \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\).
The number of favorable outcomes \(= 1\), and the total number of outcomes \(=_{26}C_{3}=2600\). So the probability \(P=\frac{1}{2600}\)

For the second problem:

Step1: Calculate the total number of ways to choose 4 fruits from \(12 + 8=20\) fruits

Using the combination formula \(C(n,r)\) with \(n = 20\) and \(r = 4\)

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Step2: Calculate the number of ways to choose 4 peaches from 12 peaches

Using the combination formula with \(n = 12\) and \(r = 4\)

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Step3: Calculate the probability

The probability \(P=\frac{_{12}C_{4}}{_{20}C_{4}}=\frac{495}{4845}=\frac{33}{323}\)

For the third problem:

Step1: Calculate the total number of ways to choose 2 teachers from \(5 + 4=9\) teachers

Using the combination formula \(C(n,r)\) with \(n=9\) and \(r = 2\)

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Step2: Calculate the number of ways to choose 2 Spanish teachers from 5 Spanish teachers

Using the combination formula with \(n = 5\) and \(r=2\)

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Step3: Calculate the probability

The probability \(P=\frac{_{5}C_{2}}{_{9}C_{2}}=\frac{10}{36}=\frac{5}{18}\)

Answer:

1. B. \(\frac{1}{2600}\)
2. \(\frac{33}{323}\)
3. D. \(\frac{_{5}C_{2}}{_{9}C_{2}}=\frac{5}{18}\)