Assume that you select 2 coins at random from 5 coins: 3 dimes and 2 quarters.
What is the probability that all of the coins selected are dimes?
To find the probability that all of the coins selected are dimes, we first need to calculate the total number of ways to choose 2 coins out of the 5 available.
We can use the combination formula to compute this. The number of combinations, denoted as C(n, r), represents the number of ways to choose r objects from a group of n objects without regard to the order in which they are chosen. The formula for the combination is:
C(n, r) = n! / (r! * (n - r)!)
Where n! represents n factorial, which is the product of all positive integers less than or equal to n.
In our case, there are 5 coins in total (3 dimes and 2 quarters), and we need to choose 2 dimes. Therefore, n = 5 and r = 2. Let's calculate the combination:
C(5, 2) = 5! / (2! * (5 - 2)!) = 5! / (2! * 3!) = (5 * 4 * 3!) / (2! * 3!) = (5 * 4) / (2 * 1) = 10
So, there are 10 possible ways to choose 2 coins out of the 5 available.
Next, we need to determine the number of ways to select 2 dimes from the 3 available dimes. Again, we can use the combination formula:
C(3, 2) = 3! / (2! * (3 - 2)!) = 3! / (2! * 1!) = (3 * 2 * 1!) / (2 * 1) = 3
So, there are 3 possible ways to select 2 dimes out of the 3 available.
Finally, we can compute the probability by dividing the number of favorable outcomes (selecting 2 dimes) by the total number of possible outcomes (selecting 2 coins):
Probability = number of favorable outcomes / total number of possible outcomes
Probability = 3 / 10
Probability = 0.3 or 30% (expressed as a decimal or percentage)
Therefore, the probability that all of the coins selected are dimes is 0.3 or 30%.
To find the probability, we need to divide the favorable outcomes (selecting all dimes) by the total possible outcomes (selecting any 2 coins).
First, let's determine the total possible outcomes. We have 5 coins in total, so the total possible outcomes can be calculated using the combination formula:
nCr = n! / (r! * (n-r)!)
In this case, we have 5 coins and need to select 2. Plugging those values into the formula, we get:
5C2 = 5! / (2! * (5-2)!)
= 5! / (2! * 3!)
= (5 * 4 * 3!) / (2! * 3!)
= (5 * 4) / 2
= 10
So, there are 10 total possible outcomes when selecting 2 coins from 5.
Next, we need to determine the favorable outcome, which is selecting 2 dimes from the 3 dimes available. We can calculate this using the combination formula as well:
3C2 = 3! / (2! * (3-2)!)
= 3! / (2! * 1!)
= 3
Therefore, there are 3 favorable outcomes when selecting 2 dimes from the 3 available.
Lastly, we can calculate the probability by dividing the favorable outcomes by the total possible outcomes:
Probability = favorable outcomes / total possible outcomes
= 3/10
So, the probability that all of the coins selected are dimes is 3/10 or 0.3.