Tasha has a piggy bank that contains 15 loonies, 30 quarters, 20 dimes, and 7 nickels.
a) if she chooses one con at a random to buy some candy, what is the probability of choosing a loonie or quarter?
b) if she chooses three coins, all at once , what is the probability of choosing 2 loonies and quarter?
There ar 72 coins total
a) (15+30)/72
b) the possible ways of choosing 3 coins is 72-choose-3 = 72!/3!(72-3)! = 70*71*72/6. This is the denominator.
There are 15-choose-2 ways of picking the two loonies and 30-choose-1 ways of picking the quarter. So, the numerator is ((14*15)/2))*30
To calculate the probabilities in both scenarios, we need to determine the total number of coins and the number of favorable outcomes.
a) Finding the probability of choosing a loonie or a quarter:
1. Calculate the total number of coins: 15 loonies + 30 quarters + 20 dimes + 7 nickels = 72 coins.
2. Determine the number of favorable outcomes: Since Tasha wants either a loonie or a quarter, there are 15 loonies + 30 quarters = 45 favorable outcomes.
3. Use the formula for probability: Probability = (Number of Favorable Outcomes) / (Total Number of Outcomes).
Probability = 45 / 72.
Simplifying the fraction, we get approximately 0.625 or 62.5%.
b) Finding the probability of choosing 2 loonies and a quarter:
1. Calculate the total number of coins: 15 loonies + 30 quarters + 20 dimes + 7 nickels = 72 coins.
2. Determine the number of favorable outcomes: To choose 2 loonies and a quarter, you need to select 2 loonies from the 15 available and 1 quarter from the 30 available.
Favorable outcomes = (Number of ways to choose 2 loonies from 15) * (Number of ways to choose 1 quarter from 30).
Favorable outcomes = (15 choose 2) * (30 choose 1).
Using combinatorial notation, this is written as C(15,2) * C(30,1).
Solve each combination: C(15,2) = (15!)/(2!(15-2)!) = (15 * 14) / (2 * 1) = 105, and C(30,1) = (30!)/(1!(30-1)!) = 30.
Therefore, favorable outcomes = 105 * 30 = 3150.
3. Use the formula for probability: Probability = (Number of Favorable Outcomes) / (Total Number of Outcomes).
Probability = 3150 / 72C3.
Simplifying the fraction, we get approximately 0.194 or 19.4%.