A jar contains five red balls numbered 1 to 5 and seven...

A jar contains five red balls numbered 1 to 5 and seven green balls numbered 1 to 7 . Two balls are drawn at random from the jar. Find the following conditional probabilities. a. The second ball drawn is red, given that the first is red. b. The second ball drawn is red, given that the first is green. c. The second ball drawn is even-numbered, given that the first is odd-numbered. d. The second ball drawn is even-numbered, given that the first is even-numbered.

A jar contains five red balls numbered 1 to 5 and seven green balls numbered 1 to 7 .
Two balls are drawn at random from the jar. Find the following conditional probabilities.
a. The second ball drawn is red, given that the first is red.
b. The second ball drawn is red, given that the first is green.
c. The second ball drawn is even-numbered, given that the first is odd-numbered.
d. The second ball drawn is even-numbered, given that the first is even-numbered.

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Key Concepts

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It involves updating the probability based on new information, using the formula P(A|B) = P(A?B) / P(B), where the denominator reflects the reduced sample space after event B has occurred. This concept is fundamental in scenarios where events are interrelated.

Sampling Without Replacement

Sampling without replacement describes the process of drawing items from a finite set in a way that each drawn item is not returned to the set before subsequent draws. This method causes the probabilities to change over time because the total number of items decreases with each draw, making successive events dependent on previous outcomes.

Combinatorial Analysis

Combinatorial analysis applies counting techniques such as permutations and combinations to determine the total number of possible outcomes and favorable cases in probability problems. This approach is essential for calculating probabilities accurately, particularly when outcomes depend on specific arrangements or selections within a finite sample space.

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