An urn has fifteen red balls and nineteen white balls. Three balls are selected at random and the number of red balls are observed.
Find the probability distribution for
Number of Red Balls:
0 =
1 =
2 =
3 =
total amount of balls = 15 red + 19 white = 34 balls
0 - 0/34 = 0
1 - 1/34
2 - 2/34 = 1/17
3 - 3/34
Prob red = 15/34
prob white = 19/34
when picking three:
prob(0 red) = C(3,0) (19/34)(18/33)(17/32) = 57/352
prob(1 red) = C(3,1) (15/34)(19/33)(18/32) = 2565/5984
prob(2red) = C(3,2) (15/34)(14/33)(19/32) = 1998/5984
prob (3red) = C(3,3)(15/34)(14/33)(13/32) = 455/5984
notice that the sum of the 4 fractions above is 5984/5984 = 1
the 2nd last prob calculation should be 1995/5984
To find the probability distribution for the number of red balls, we can use the concept of combinations and probability. Here's how you can get the answer:
Step 1: Determine the total number of ways to select 3 balls from a total of 15 red balls and 19 white balls.
The total number of ways to select 3 balls from a total of 34 balls can be calculated using combinations. The formula for combinations is:
C(n, r) = n! / [(n-r)! * r!]
Applying this formula:
C(34, 3) = 34! / [(34-3)! * 3!]
= (34 * 33 * 32) / (3 * 2 * 1)
= 5984
So, there are a total of 5984 ways to select 3 balls from the urn.
Step 2: Calculate the probability of getting 0 red balls.
Since there are 15 red balls in the urn, the probability of not selecting any red balls can be calculated by dividing the number of ways to select 3 balls from the 19 white balls only by the total number of ways to select 3 balls from the urn.
C(19, 3) = 19! / [(19-3)! * 3!]
= (19 * 18 * 17) / (3 * 2 * 1)
= 969
P(0 red balls) = C(19, 3) / C(34, 3)
= 969 / 5984
≈ 0.162
So, the probability of getting 0 red balls is approximately 0.162.
Step 3: Calculate the probability of getting 1 red ball.
The probability of getting 1 red ball can be calculated by dividing the number of ways to select 1 red ball and 2 white balls by the total number of ways to select 3 balls from the urn.
C(15, 1) = 15! / [(15-1)! * 1!]
= (15) / (1)
= 15
C(19, 2) = 19! / [(19-2)! * 2!]
= (19 * 18) / (2 * 1)
= 171
P(1 red ball) = (C(15, 1) * C(19, 2)) / C(34, 3)
= (15 * 171) / 5984
≈ 0.438
So, the probability of getting 1 red ball is approximately 0.438.
Step 4: Calculate the probability of getting 2 red balls.
The probability of getting 2 red balls can be calculated by dividing the number of ways to select 2 red balls and 1 white ball by the total number of ways to select 3 balls from the urn.
C(15, 2) = 15! / [(15-2)! * 2!]
= (15 * 14) / (2 * 1)
= 105
C(19, 1) = 19! / [(19-1)! * 1!]
= (19) / (1)
= 19
P(2 red balls) = (C(15, 2) * C(19, 1)) / C(34, 3)
= (105 * 19) / 5984
≈ 0.52
So, the probability of getting 2 red balls is approximately 0.52.
Step 5: Calculate the probability of getting 3 red balls.
The probability of getting 3 red balls can be calculated by dividing the number of ways to select 3 red balls by the total number of ways to select 3 balls from the urn.
C(15, 3) = 15! / [(15-3)! * 3!]
= (15 * 14 * 13) / (3 * 2 * 1)
= 455
P(3 red balls) = C(15, 3) / C(34, 3)
= 455 / 5984
≈ 0.076
So, the probability of getting 3 red balls is approximately 0.076.
Therefore, the probability distribution for the number of red balls is:
0 = 0.162
1 = 0.438
2 = 0.52
3 = 0.076