An urn contains four red balls and three white balls. A sample of 2 balls is selected at random from the urn. What is the probability that at least one of the balls is red?
number of ways to pick any two = C(7,2) = 21
number of ways with only white = C(3,2) = 3
so the number of ways with at least one red = 21-3 = 18
prob of at least 1 red = 18/21 = 6/7
what does c stand for and how do you get 21?
To find the probability that at least one of the balls drawn is red, we need to consider two scenarios:
Scenario 1: One red ball and one white ball are drawn
Scenario 2: Two red balls are drawn
Let's calculate the probability for each scenario and then sum them up.
Scenario 1:
The probability of drawing one red ball and one white ball can be calculated as:
Probability = (Number of ways to choose 1 red ball out of 4) multiplied by (Number of ways to choose 1 white ball out of 3)
divided by (Total number of ways to choose 2 balls out of 7)
Probability = (4 choose 1) * (3 choose 1) / (7 choose 2)
Scenario 2:
The probability of drawing two red balls can be calculated as:
Probability = (Number of ways to choose 2 red balls out of 4) divided by (Total number of ways to choose 2 balls out of 7)
Probability = (4 choose 2) / (7 choose 2)
Now, let's calculate the probabilities:
Probability of scenario 1:
(4 choose 1) = 4
(3 choose 1) = 3
(7 choose 2) = 21
Probability of scenario 1 = (4 * 3) / 21 = 12 / 21 = 4 / 7
Probability of scenario 2:
(4 choose 2) = 6
(7 choose 2) = 21
Probability of scenario 2 = 6 / 21 = 2 / 7
Now, let's sum up the probabilities of the two scenarios:
Total probability = Probability of scenario 1 + Probability of scenario 2
= (4 / 7) + (2 / 7)
= 6 / 7
Therefore, the probability that at least one of the balls drawn is red is 6/7 or approximately 0.857.
To find the probability that at least one of the balls is red, you can calculate the probability of the event happening and subtract it from 1 to find the complementary event (which means at least one ball is red).
Step 1: Find the total number of possible outcomes.
Since we are selecting 2 balls from the urn, the total number of possible outcomes is the number of ways we can pick 2 balls from a total of 7 balls. This can be calculated using the combination formula: C(n, r) = n! / (r!(n-r)!), where n is the total number of objects and r is the number of objects chosen. In this case, n = 7 and r = 2.
C(7, 2) = 7! / (2!(7-2)!)
= 7! / (2!5!)
= (7 * 6 * 5!) / (2! * 5!)
= (7 * 6) / 2!
= 42 / 2
= 21
So, there are 21 possible outcomes when selecting 2 balls from the urn.
Step 2: Find the number of desired outcomes.
The desired outcome is when at least one of the balls is red. There are two possibilities for this: either one red ball and one white ball are selected, or both balls are red. We need to calculate the number of ways these two possibilities can occur.
Possibility 1: One red ball and one white ball are selected.
The number of ways to choose 1 red ball from the 4 red balls is C(4, 1) = 4, and the number of ways to choose 1 white ball from the 3 white balls is C(3, 1) = 3. Since these are independent events, we can multiply these numbers together to find the total number of ways one red ball and one white ball can be selected.
Number of ways = C(4, 1) * C(3, 1) = 4 * 3 = 12
Possibility 2: Both balls are red.
The number of ways to choose 2 red balls from the 4 red balls is C(4, 2) = 6.
Number of ways = C(4, 2) = 6
So, the total number of desired outcomes is 12 + 6 = 18.
Step 3: Calculate the probability.
The probability of an event happening is given by the formula P(event) = number of desired outcomes / total number of possible outcomes.
P(at least one red ball) = number of desired outcomes / total number of possible outcomes
= 18 / 21
Therefore, the probability that at least one of the balls selected is red is 18/21, which can be simplified as 6/7 or approximately 0.857 (rounded to three decimal places).