Four balls are selected at random without replacement from an urn containing four white balls and five blue balls. Find the probability of the given event.
All of the balls are blue.
To find the probability of all four balls being blue, we'll first calculate the total number of ways to select four balls from the urn, and then find the number of ways to select four blue balls.
Step 1: Calculating the total number of ways to select four balls from the urn:
The total number of balls in the urn is 4 white balls + 5 blue balls = 9 balls.
Therefore, the total number of ways to select 4 balls from 9 is given by the combination formula:
C(9, 4) = 9! / (4! * (9-4)!) = 9! / (4! * 5!) = (9 * 8 * 7 * 6) / (4 * 3 * 2 * 1) = 126.
So, there are 126 ways to select four balls from the urn.
Step 2: Calculating the number of ways to select four blue balls:
Since there are 5 blue balls in the urn, the number of ways to select 4 blue balls can be calculated using the same combination formula:
C(5, 4) = 5! / (4! * (5-4)!) = 5! / (4! * 1!) = 5 / 1 = 5.
So, there are 5 ways to select four blue balls.
Step 3: Calculating the probability:
The probability of an event is given by the number of favorable outcomes divided by the total number of possible outcomes.
Therefore, the probability of all four balls being blue is:
P(all blue) = Number of ways to select four blue balls / Total number of ways to select four balls
= 5 / 126.
Hence, the probability of all four balls being blue is 5/126.
To find the probability that all of the balls selected are blue, we need to determine the total number of possible outcomes and the number of favorable outcomes.
First, let's determine the total number of possible outcomes. Since we are selecting four balls without replacement, the total number of possible outcomes can be calculated using combinations.
There are a total of nine balls in the urn: four white balls and five blue balls. We need to select four balls, so the total number of outcomes is represented by C(9, 4), which is calculated as follows:
C(9, 4) = 9! / (4! * (9-4)!) = 9! / (4! * 5!) = (9 * 8 * 7 * 6) / (4 * 3 * 2 * 1) = 126
Therefore, there are 126 possible outcomes when selecting four balls from the nine available.
Now, let's determine the number of favorable outcomes, which is the number of ways to select four blue balls from the five available. This can also be calculated using combinations.
The number of favorable outcomes is represented by C(5, 4), which is calculated as follows:
C(5, 4) = 5! / (4! * (5-4)!) = 5! / (4! * 1!) = (5 * 4!) / (4 * 3 * 2 * 1) = 5
Therefore, there are 5 favorable outcomes when selecting four blue balls from the five available.
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = (Number of Favorable Outcomes) / (Number of Total Possible Outcomes)
Probability = 5 / 126
Probability ≈ 0.0397
Thus, the probability of selecting four blue balls from the urn is approximately 0.0397, or 3.97%.
Right
4 w and 5 b so 9
first one
5/9
second
4/8
third
3/7
fourth
2/6
so
5/9 * 1/2 * 3/7 * 1/3
(5)/(7*18) = 5/126