Conditional probability? There is a bag with 10 marbles in it; 8 are black, 2 white

[Katlyn]

I hope someone can help! I can’t solve this at all ?

• There is a bag with 10 marbles in it
• 8 are black
• 2 are white
• You pick a marble
• If it is black you pick again
• Once you pick a white marble you stop picking
• What is the probability that after a maximum of 4 picks you will have chosen a white marble?

I can get my head to working out the probability of choosing a white marble each round, but not to what the overall probability that a white will be chosen over 4 rounds! Although, I may well be wrong so don’t assume I’ve understood that bit correctly ?

It would be great if someone could explain to me how to solve this as well as what the correct answer is! I want to understand, not just be given the answer

Thank you for any help!

 

[tkhunny]

I hope someone can help! I can’t solve this at all ?

• There is a bag with 10 marbles in it
• 8 are black
• 2 are white
• You pick a marble
• If it is black you pick again
• Once you pick a white marble you stop picking
• What is the probability that after a maximum of 4 picks you will have chosen a white marble?

I can get my head to working out the probability of choosing a white marble each round, but not to what the overall probability that a white will be chosen over 4 rounds! Although, I may well be wrong so don’t assume I’ve understood that bit correctly ?

It would be great if someone could explain to me how to solve this as well as what the correct answer is! I want to understand, not just be given the answer

Thank you for any help!

Why not just build a tree?

Round1

8/10 Black vs 2/10 White -- 20% chance that you will have only one draw.

80% chance that you continue.

Round2

7/9 Black vs 2/9 White -- 22.22% chance that you stop with two draws.

77.78% chance that you continue.

Round3 - You go!

 

[Katlyn]

Why not just build a tree?

Round1

8/10 Black vs 2/10 White -- 20% chance that you will have only one draw.

80% chance that you continue.

Round2

7/9 Black vs 2/9 White -- 22.22% chance that you stop with two draws.

77.78% chance that you continue.

Round3 - You go!

Thank you for your reply

Sorry, I forgot to clarify that you put the black marble back each time you select again. Would this mean that it’s a 20% chance of getting a white, regardless of how many iterations you repeat? Somehow that doesn’t seem to make sense in my head; I know you’d have a 20% chance each selection...but does that equate to the overall chance of getting a white, whether you picked 4 times or 50 times?? Surely you’d be more likely to pick a white if you did it 50 times than if you did it 4 times??

 

[tkhunny]

Thank you for your reply

Sorry, I forgot to clarify that you put the black marble back each time you select again. Would this mean that it’s a 20% chance of getting a white, regardless of how many iterations you repeat? Somehow that doesn’t seem to make sense in my head; I know you’d have a 20% chance each selection...but does that equate to the overall chance of getting a white, whether you picked 4 times or 50 times?? Surely you’d be more likely to pick a white if you did it 50 times than if you did it 4 times??

Yes please specify with or without replacement. It is important with such a small population.

Don't be afraid of it. The fact that p(50 black in a row) > 0 is no concern for your 4-draw problem.