# Probability question for a genius!?

A speaks truth 2 out of 3 times and B speaks truth 4 out of 5 times. They agree in assertion that from a bag containing 6 balls of different colors, a red ball has been drawn. Find the probability that the statement is true.

Relevance

I believe 8/13 is correct.

Let R = event that ball is indeed red (ie statement is true).

S = event that both A and B independently say the ball is red

We need to find P(R | S)

= P(R n S) / P(S)

= P(R n S) / [P(R n S) + P(R' n S)] ... (1)

R n S means that the ball is red and A and B were telling the truth

P(R n S) = 1/6 * 2/3 * 4/5 = 8/90

R' n S means that the ball is not red and A and B were lying

P(R' n S) = 5/6 * 1/3 * 1/5 = 5/90

From (1), P(R | S) = 8/13

• You coud get the statement "a red ball has been drawn" in the following two ways:

Case 1:

A red ball was drawn (1/6) and both A and B told the truth (2/3 * 4/5). All together this is 8/90

Case 2:

A non-red ball was drawn (5/6) and both A and B told a lie (1/3 * 1/5). All together this is 5/90

So the chance you have fallen into case 1 is:

8/90 / (8/90 + 5/90)

= 8/13

• Since the question is based on the assertions of the two, then the probability distribution of the bag is moot. (Either a red ball is drawn or isn't.) Thus the probability of the validity of the red ball drawn is contingent on both speaking the truth. Assuming that the two assertions are independent of each other, probability law indicates P(both) = P(A) * P(B). As P(A) = 2/3 and P(B) = 4/5, then the product is 8/15. This is the probability that the statement is true.

However, if they are in collaboration, then the greater probability dominates over the other, in which case the probability is 4/5.

Source(s): My early morning analysis.
• The 1/6 probability of choosing a red ball is useless information since you are told that a red ball has already been chosen

let

A = "A tells truth"

B = "B tells truth"

The honesty of A & B is independent, so

P(A∩B)

= P(A) P(B)

= (2/3)(4/5)

= 8/15

If they are in collaboration then the probability depends on which of A or B has the biggest influence, it can't be assumed that the most honest one will make the final decision.

,.,,..,

• I believe the answer is Bayes. P (A|B) . Thus each respondent(A & B) is answering Red = positive. Each answer is similar to two drug tests, which are either accurate or delivering false positives. My answer :

(1/6 * 2/3) / [1/6 * 2/3 + 5/6 * 1/3] = .2857 P of A is right

(1/6 * 4/5) / ]1/6 * 4/5 + 5/6 * 1/5] = .4443 P of B is right

Both must be right, so : P Red = Positive is: .2857 * .4443 = .1269 or approx 12.7%

• In my opinion this problem is insolvable, by the information given. This is why:

In order to find the answer you need to turn the question round a bit:

How likable is it that a different coloured ball is pulled and both (A + B) tell A lie about it?

That's 5/6 * 1/3 * 1/5 = 5.55%

So the probability of the statement being true would be the opposite, which is 94.45%

However, that's assuming that they're making their decision completely independent from each other (saying they don't know each other, not in the same room, etc...) which doesn't seem to be the case, as we're told that they AGREE on it being red - you would have to do some hundred years of research on how people from diffrent cultures and ages act in situations like, just to find a statistical probability that's still gonna be a statistic, which is worth ****.

Another problem:

The probability of them choosing the same color for their lie depends, like someone mentioned already, on what lies they're "allowed" to tell. Since we're not told about that, we have to assume they can say whatever they want. Which brings the possability of them choosing the same lie pretty much down to zero, and therefor rises the probability of the statement being true up to....something like 99.9999999% (let's say one of them is a painter and and show of....).

So yeah.... not enough information I'd say. I always hated probability questions which involve human minds, just doesn't make sense to me. But hey, if you can prove me wrong I'm ready to learn...

• The field is infinite. How many workers are there in total? 10/x = percentage of workers chosen. As 1/2 commuters ride train, 1/2 get to work in other ways. Random sample of 10 workers could produce all walkers/all train riders, etc. It's not simply 50%*10=5. Probability says within 95% confidence, it's likely 5 of 10 are train riders, however, as this is the MOST likely outcome given the parameters.

• probability of a speaks truth is 2/3

probability of b speaks truth is 4/5

probability of ball to be red is 1/6

probability if a speaks red and is red is 1/9

probability of a speaks lies that ball is red is 1/18

probability of b speaks truth that ball is red is 2/15

probability of b speaks lies is 1/30

probability that both a and b speaks truth that ball is red is 4/90

probability that both speaks lies that ball is red is 1/90

• There is a 73.34% chance that the ball is red, or 1 in 1.36 or 7.3/10

Although I only speak the truth 1 out of 3 times, so the above statement is 33.33% likely to be true or 1 in 3

• All the factors are independent, so you multiply.

Colour: 1/6

A : 2/3

B : 4/5

Truth: 1/6 * 2/3 * 4/5 = 8/ 90 = 4/ 45