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Raina places the following balls into a bag. She then draws three balls out of the bag, one at a time, without replacement. What is the probability that the colors of the balls alternate?
 

There are 3 oragne balls and 5 purple balls. 

 

I have figured out what the probablilities of the ballsg getting alternated. 

 

If she picked orange first, it would be 3/7 x 4/6 x 2/5 = 24/210

If she picked purple first, it would be 5/7 x 4/6 x 4/5 = 80/210 

 

What should I use after this? How can I come to a conclusion? 

 

Thanks in advance. 

 Mar 21, 2024
 #1
avatar+179 
+1

The probability of Raina getting alternating colors depends on whether she picks an orange ball or a purple ball first. Let's analyze both scenarios:

 

Scenario 1: Orange - Purple - Orange (OPO)

 

Probability of picking an orange ball first: 3 out of 8 total balls (3 orange + 5 purple).

 

Probability of picking a purple ball after an orange: 5 out of 7 remaining balls (1 orange left + 5 purple).

 

Probability of picking another orange ball after a purple: 2 out of 6 remaining balls (2 orange left).

 

So, the probability of OPO is (3/8) * (5/7) * (2/6) = 5/56.

 

Scenario 2: Purple - Orange - Purple (PPO)

 

Probability of picking a purple ball first: 5 out of 8 total balls.

 

Probability of picking an orange ball after a purple: 3 out of 7 remaining balls (3 orange left + 4 purple).

 

Probability of picking another purple ball after an orange: 4 out of 6 remaining balls (4 purple left).

 

So, the probability of PPO is (5/8) * (3/7) * (4/6) = 5/56.

 

Total Probability

 

Since these scenarios (OPO and PPO) are mutually exclusive (they can't happen at the same time), to get the overall probability, we simply add the probabilities of each scenario:

 

Total Probability = Probability(OPO) + Probability(PPO) = (5/56) + (5/56) = 10/56 = 5/28

 

Therefore, the probability of Raina drawing three balls with alternating colors (orange-purple-orange or purple-orange-purple) is 5/28.

 Mar 21, 2024
 #2
avatar-487 
+1

The probability of Raina picking alternating colors can be calculated by considering the two possibilities for the first ball she draws: orange or purple.

 

Scenario 1: Orange first (Orange - Purple - Orange)

 

Probability of picking an orange ball first: 3 out of 8 total balls (since there are 3 orange balls and 8 total at the beginning).

 

Probability of picking a purple ball second: 5 out of 7 remaining balls (after taking out the orange ball, there are 7 left, and 5 of them are purple).

 

Probability of picking an orange ball third: 2 out of 6 remaining balls (after taking out the purple ball, there are 6 left, and 2 of them are orange).

 

So, the probability for this scenario is (3/8) * (5/7) * (2/6) = 5/56.

 

Scenario 2: Purple first (Purple - Orange - Purple)

 

Similarly, we can find the probability for picking purple first, then orange, and then purple again.

 

Probability of picking a purple ball first: 5 out of 8 total balls.

 

Probability of picking an orange ball second: 3 out of 7 remaining balls.

 

Probability of picking a purple ball third: 4 out of 6 remaining balls.

 

The probability for this scenario is (5/8) * (3/7) * (4/6) = 5/56.

 

Total Probability

 

Since these two scenarios (orange first and purple first) are mutually exclusive (they can't happen at the same time), to get the total probability of picking alternating colors, we simply add the probabilities of each scenario:

 

Total Probability = (5/56) + (5/56) = 10/56 = 5/28

 

Therefore, the probability of Raina picking alternating colored balls is 5/28.

 Mar 21, 2024
 #3
avatar+128796 
+1

P  O   P  =   (5/8) (3/7) ( 4/6)  = 5 / 28  =  10/56

O  P   O  =  (3/8) (5/7) ( 2/6)  = 5 / 56  =    5/56

 

15 / 56 

 

 

cool cool cool 

 Mar 21, 2024

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