Conditional Probability

According to given informations, the required probability

Let be the event of obtained sum of and be the event of obtained sum of .

Now,
and

Neither a sum of nor a sum of shows of


Required probability

Step 1: Understanding the Problem We are tasked with finding the probability that exactly 5 of the balls in the bag are green given that 3 randomly drawn balls are all green.
Step 2: Applying Bayes' Theorem Let the events be defined as follows: - = "Exactly 5 of the 6 balls are green." - = "Three balls drawn are all green." We need to calculate , the probability that there are exactly 5 green balls given that 3 drawn balls are green. By Bayes' Theorem:
Step 3: Calculating Each Probability
Step 3a: Probability of (Probability of exactly 5 green balls in the bag) Since 5 out of 6 balls are green:
Step 3b: Probability of (Probability of drawing 3 green balls if there are exactly 5 green balls) If there are 5 green balls in the bag, the total ways to choose 3 balls that are all green: - Choose 3 balls from 5 green balls: - Choose any 3 balls out of 6 total balls: Thus,
Step 3c: Probability of (Total probability that 3 drawn balls are green) This includes two scenarios: 1. 5 green balls in the bag Probability = and probability of drawing 3 green balls = 2. 6 green balls in the bag Probability = and probability of drawing 3 green balls = 1 By total probability law:
Step 4: Calculate By Bayes' theorem:
Step 5: Final Answer