You have three toonies and two loonies in your pocket. You pull one coin out and then pull out another (without replacing the first)
1) show the total of all the probabilities is 1
2) show using combinations that part 1) is correct
Cases:
LL -- (2/5)(1/4) = 1/10
TT -- (3/5)(2/4) = 3/10
LT -- (2/5)(3/4) = 3/10
TL -- (3/5)(2/4) = 3/10
SUM = 1/10 + 3/10 + 3/10 + 3/10
= 10/10
= 1
in combination format
choose 2 of the 5 coins = C(5,2) = 10
Two loonies = C(2,2) = 1
Two toonies = C(3,2) = 3
one of each = C(3,1)xC(2,1) = 6
total probs = 1/10 + 3/10 + 6/10 = 1
btw, "loonie" - nickname for Canadian $1 coin, it has a loon on it.
"toonie" - nickname for Canadian $2 coin
To solve this problem, let's first determine the total number of possible outcomes.
Step 1: Total Number of Coins
We have 5 coins in total - 3 toonies and 2 loonies.
Step 2: Selecting the First Coin
We can choose any of the 5 coins as the first coin, so there are 5 possibilities.
Step 3: Selecting the Second Coin
Once we have already selected the first coin, there are only 4 remaining coins to choose from.
Step 4: Calculating the Total Number of Outcomes
By multiplying the number of possibilities from steps 2 and 3, we get the total number of outcomes:
Total Outcomes = 5 * 4 = 20
Now let's solve the questions.
1) Showing that the total of all the probabilities is 1:
We can calculate the probability of each outcome and sum them up to verify that the total probability is 1.
Probability of choosing a toonie first: 3/5 (since there are 3 toonies out of 5 coins)
Probability of choosing a loonie first: 2/5 (since there are 2 loonies out of 5 coins)
If we select a toonie first, the probability of selecting a loonie second is: 2/4 (as there are now 4 coins left)
If we select a loonie first, the probability of selecting a toonie second is: 3/4 (as there are now 4 coins left)
Now we can sum up the probabilities:
(3/5) * (2/4) + (2/5) * (3/4) = 6/20 + 6/20 = 12/20 = 3/5
Therefore, the total probability is 3/5, which is 1 when expressed as a decimal (1/1 = 5/5, so 3/5 represents 3 out of the 5 parts).
2) Showing using combinations that part 1) is correct:
Using combinations, we can calculate the number of ways to select a toonie and a loonie without regard to the order of selection.
The number of ways to select a toonie: C(3, 1) = 3 (choosing 1 toonie from a set of 3)
The number of ways to select a loonie: C(2, 1) = 2 (choosing 1 loonie from a set of 2)
To get the total number of outcomes, we multiply the choices from both steps:
Total Outcomes = C(3, 1) * C(2, 1) = 3 * 2 = 6
This demonstrates that there are 6 possible outcomes, consistent with our calculation in step 4.
In conclusion, we have shown in two different ways (summing probabilities and using combinations) that the total number of outcomes is 6, which confirms that the total of all probabilities is indeed 1.
To answer both parts of your question, let's break it down step by step:
Step 1: Identify the possible outcomes
In this scenario, there are five possible outcomes when pulling out two coins: TT, TL, LT, LL, and TL. T represents a toonie, and L represents a loonie.
Step 2: Calculate the probability of each outcome
Since we have three toonies and two loonies, the probability of pulling out either coin at the start is:
P(T) = 3/5 (three toonies out of five coins)
P(L) = 2/5 (two loonies out of five coins)
For the first outcome (TT), let's calculate its probability:
P(TT) = P(T) * P(T|T)
The probability of pulling the second toonie given that the first coin was a toonie is P(T|T) = 2/4 (two remaining toonies out of four remaining coins). Therefore:
P(TT) = (3/5) * (2/4) = 6/20 = 3/10
Similarly, calculate the probabilities for the other outcomes:
P(TL) = P(T) * P(L|T) = (3/5) * (2/4) = 6/20 = 3/10
P(LT) = P(L) * P(T|L) = (2/5) * (3/4) = 6/20 = 3/10
P(LL) = P(L) * P(L|L) = (2/5) * (1/4) = 2/20 = 1/10
Step 3: Calculate the total probability
To calculate the total probability, sum up the probabilities of all possible outcomes:
P(TT) + P(TL) + P(LT) + P(LL) = (3/10) + (3/10) + (3/10) + (1/10) = 10/10 = 1
Therefore, we have shown that the total probability of all outcomes is 1.
Now, let's move on to the second part of your question, using combinations to demonstrate the correctness of part 1.
Step 4: Use combinations to show the correctness
We can further support the correctness of part 1 by using combinations to count the number of possible outcomes.
Total possible outcomes when pulling two coins out without replacement = C(5, 2)
To calculate C(5, 2), we use the combination formula:
C(n, r) = n! / (r! * (n-r)!)
C(5, 2) = 5! / (2! * (5-2)!) = 5! / (2! * 3!) = (5 * 4 * 3!) / (2! * 3!) = (5 * 4) / 2! = 10
So, there are 10 possible outcomes when pulling two coins out of the pocket.
Step 5: Calculate the probabilities using combinations
Using combinations, we can consider each possible outcome to determine its probability.
Number of outcomes with two toonies (TT) = C(3, 2) = 3! / (2! * (3-2)!) = 3
Number of outcomes with one toonie and one loonie (TL and LT) = C(3, 1) * C(2, 1) = 3 * 2 = 6
Number of outcomes with two loonies (LL) = C(2, 2) = 1
Total number of outcomes = 3 + 6 + 1 = 10
Now we can compare the probabilities calculated using combinations to those we found earlier using conditional probabilities:
P(TT) = 3/10, P(TL) = 3/10, P(LT) = 3/10, P(LL) = 1/10
Since the probabilities calculated using combinations match with the ones we previously obtained using conditional probabilities, this further confirms the correctness of part 1.
In conclusion, both calculations demonstrate that the total probability of all outcomes is 1, and combinations further validate this result.