A box contains 5 red balls and 4 green balls. Two balls are drawn one after another without replacement. What is the probability that the second ball is green, given that the first ball drawn was red?
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Use conditional probability formula:
\[
P(B \mid A) = \frac{P(A \cap B)}{P(A)}
\]
and adjust the total count for “without replacement.”
To determine the probability that the second ball is green, conditioned on the first ball drawn being red, we must analyze the sequence of events and their associated probabilities.
Step 1: Establish Total Initial Ball Count The box initially contains 5 red balls and 4 green balls, resulting in a total of 9 balls.
Step 2: Calculate Probability of Drawing a Red Ball First The probability of the first ball drawn being red is calculated as:
\[ P(\text{Red first}) = \frac{5}{9} \]
Step 3: Calculate Conditional Probability of Drawing a Green Ball Second Following the removal of one red ball, 8 balls remain in the box, of which 4 are green. The probability that the second ball drawn is green, given the first was red, is:
\[ P(\text{Green second | Red first}) = \frac{4}{8} = \frac{1}{2} \]
Consequently, the probability that the second ball drawn is green, given that the first ball drawn was red, is \(\frac{1}{2}\).