41 different balls are drawn from a bag containing balls numbered 1,2,…,300. The probability that the number of odd balls drawn is larger than the number of even balls can be expressed as ab where a and b are coprime positive integers. What is the value of a+b?
To find the probability that the number of odd balls drawn is larger than the number of even balls, we need to consider each possible outcome and count the ones that satisfy the condition.
Let's break down the problem into smaller steps:
Step 1: Determine the possible outcomes of drawing 41 balls from the bag.
Since there are 300 balls in the bag numbered from 1 to 300, we have a total of 300 Choose 41 possible outcomes:
C(300, 41) = 300! / (41!(300 - 41)!) = 104_591_267_047_959_729_566_085_010_939_423_344_746_257_274_373_302_171_495
Step 2: Counting the outcomes that satisfy the condition.
To count the outcomes where the number of odd balls drawn is larger than the number of even balls, we need to consider two cases:
Case 1: The number of odd balls drawn is greater than 21.
In this case, we need to count the outcomes where we draw 22 or more odd balls. We can calculate the number of outcomes as follows:
C(150, 41) + C(152, 41) + C(154, 41) + ... + C(299, 41) + C(300, 41)
Case 2: The number of odd balls drawn is equal to 21.
In this case, we need to count the outcomes where we draw exactly 21 odd balls. We can calculate the number of outcomes as follows:
C(150, 21) * C(150, 20) + C(150, 22) * C(150, 19) + C(150, 24) * C(150, 18) + ... + C(150, 40) * C(150, 1) + C(150, 41)
Step 3: Calculate the probability.
To find the probability, we divide the count of outcomes satisfying the condition by the total number of possible outcomes:
Probability = (Count of outcomes satisfying the condition) / (Total number of possible outcomes)
Step 4: Simplify the probability fraction.
To express the probability as a fraction in the form ab, where a and b are coprime positive integers, we'll simplify the fraction obtained in Step 3.
Finally, we add the numerator and denominator of the simplified fraction to get the value of a + b.