there are two urns containing colored balls . The urn A contains 5 red balls and 5 green . The urn B contains 3 red balls and 7 green balls . One of the two urns is randomly chosen ( both urns have probability 50% of being chosen ) and then a ball is drawn at random from one of the two urns .
Pr(A) = 5 , Pr(B) = 5 , Pr (G|A) = 5 , Pr ( G|B)=7
You will win if a green ball is drawn . What is the total probability of your winning in this set-up ?
To determine the total probability of winning in this setup, we need to calculate the probability of winning when each urn is chosen, and then sum up those probabilities.
Let's break down the problem step by step:
Step 1: Probability of choosing urn A or B
Since both urns have a 50% chance of being chosen, the probability of choosing urn A, denoted as Pr(A), is 0.5, and the probability of choosing urn B, denoted as Pr(B), is also 0.5.
Step 2: Probability of winning given that urn A is chosen
When urn A is chosen, there are 5 red balls and 5 green balls. Since the objective is to win by drawing a green ball, the probability of winning given that urn A is chosen, denoted as Pr(W|A), is the probability of drawing a green ball from urn A.
Pr(G|A) = 5/10 = 0.5
Step 3: Probability of winning given that urn B is chosen
When urn B is chosen, there are 3 red balls and 7 green balls. Again, the objective is to win by drawing a green ball, so the probability of winning given that urn B is chosen, denoted as Pr(W|B), is the probability of drawing a green ball from urn B.
Pr(G|B) = 7/10 = 0.7
Step 4: Calculating the total probability of winning
To calculate the total probability of winning, we need to consider the probability of winning given each possible scenario, weighted by the probability of each scenario occurring.
Total Probability of Winning = Pr(A) * Pr(W|A) + Pr(B) * Pr(W|B)
= 0.5 * 0.5 + 0.5 * 0.7
= 0.25 + 0.35
= 0.6
Therefore, the total probability of winning in this setup is 0.6 or 60%.