In a randomly generated sequence of 24 binary digits (0s and 1s), what is the probability that exactly half of the digits are 0?
A. 1352078
B. 1.295
C. 0
D. 0.1612
Answer (1)
Calculate the total possible sequences: 2 24 = 16777216 .
Calculate the number of sequences with 12 zeros and 12 ones: ( 12 24 ) = 2704156 .
Calculate the probability: 16777216 2704156 .
The probability that exactly half of the digits are 0 is approximately: 0.1612 .
Explanation
Understand the problem We are given a sequence of 24 binary digits (0s and 1s). We want to find the probability that exactly half of the digits are 0. This means we want to find the probability of having exactly 12 zeros and 12 ones in the sequence.
Calculate the total number of sequences First, we need to calculate the total number of possible sequences of 24 binary digits. Since each digit can be either 0 or 1, there are 2 24 possible sequences.
Calculate the number of sequences with 12 zeros and 12 ones Next, we need to calculate the number of sequences with exactly 12 zeros and 12 ones. This is a binomial coefficient, which can be calculated as ( 12 24 ) = 12 ! 12 ! 24 ! . The result of this calculation is 2704156.
Calculate the probability Now, we calculate the probability by dividing the number of sequences with 12 zeros and 12 ones by the total number of possible sequences: P = 2 24 ( 12 24 ) = 16777216 2704156 .
Approximate the probability and conclude Finally, we approximate the value of the probability: P = 16777216 2704156 ≈ 0.1612 . Therefore, the probability that exactly half of the digits are 0 is approximately 0.1612.
Examples
Consider a scenario where you're flipping a fair coin 24 times. The probability of getting exactly 12 heads (or 12 tails) can be calculated using the same approach. This is because each coin flip is a binary event (heads or tails), and we want to find the probability of a specific combination (12 heads and 12 tails). The formula 2 n ( k n ) gives the probability of getting exactly k successes in n trials, where each trial has two equally likely outcomes.
