A box contains 10 blue, 5 yellow and 8 orange balls. Two balls are drawn at random. The probability that none of the balls is yellow is [tex]\frac{153}{253}[/tex].
To solve this problem you can use the combination and probability formulas.
The Combination Formulas:
C(n, r) = n! ÷ (n - r)! r!
C(n, r) = [tex]\frac{n!}{(n-r)!r!}[/tex]
*noted:
n = the number of items
r = how many items are taken at a time
The Probability Formulas:
p(A) = n(A) ÷ n(S)
p(A) = [tex]\frac{n(A)}{n(S)}[/tex]
*Noted:
P(A) = the probability of an event “A”
n(A) = the amount taken
n(S) = the total number of events in the sample space.
Step-by-step explanation:
Given:
- Amount of blue balls is 10 balls
- Amount of yellow balls is 5 balls
- Amount of orange balls is 8 balls
- Two balls are drawn at random.
Question:
What is the probability that none of the balls is yellow?
Solution:
Step 1
Find the number of ways to get two blue balls.
C(n, r) = n! ÷ (n - r)! r!
C(10, 2) = 10! ÷ (10 - 2)! 2!
C(10, 2) = 10! ÷ 8! 2!
C(10, 2) = (10 x 9 x 8!) ÷ (8! x 2!)
C(10, 2) = (10 x 9) ÷ (2 x 1)
C(10, 2) = 90 ÷ 2
C(10, 2) = 45
Step 2
Find the number of ways to get two orange balls.
C(n, r) = n! ÷ (n - r)! r!
C(10, 2) = 8! ÷ (8 - 2)! 2!
C(10, 2) = 8! ÷ 6! 2!
C(10, 2) = (8 x 7 x 6!) ÷ (6! x 2 x 1)
C(10, 2) = (8 x 7) ÷ (2 x 1)
C(10, 2) = 56 ÷ 2
C(10, 2) = 28
Step 3
Find the number of ways to get one blue ball and one orange ball.
C(n, r) + C(n, r)
= C(10, 1) + C(8, 1)
= (10! ÷ (10 - 1)! 1!) x (8! ÷ (8 - 1)! 1!)
= (10! ÷ 9! 1!) x (8! ÷ 7! 1!)
= ((10 x 9!) ÷ (9! x 1)) x ((8 x 7!) ÷ (7! x 1))
= (10 ÷ 1) x (8 ÷ 1)
= 10 x 8
= 80
Step 4
Find the number of ways to get two the balls that none balls is yellow.
⇔ The number of two the balls or none balls is yellow is the number of two blue balls or two orange balls or one blue balls and one orange balls
⇒ The number of two the balls or none balls is yellow = 45 + 28 + 80
⇒ The number of two the balls or none balls is yellow = 153
n(A) = 153
Step 5
Find the number of ways to get two the balls.
C(n, r) = n! ÷ (n - r)! r!
C(23, 2) = 23! ÷ (23 - 2)! 2!
C(10, 2) = 23! ÷ 21! 2!
C(10, 2) = (23 x 22 x 21!) ÷ (21! x 2!)
C(10, 2) = (23 x 22) ÷ (2 x 1)
C(10, 2) = 506 ÷ 2
C(10, 2) = 253
n(S) = 253
Step 6
Find the probability that none of the balls is yellow.
p(A) = n(A) ÷ n(S)
p(A) = [tex]\frac{n(A)}{n(S)}[/tex]
p(A) = [tex]\frac{153}{253}[/tex]
So, the probability that none of the balls is yellow is [tex]\frac{153}{253}[/tex]
Learn more about:
- A pot has 6 shiny balls and 14 matte balls. If a ball is removed from the pot without replacement: brainly.ph/question/29163450
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