Question

What is the probabilty that exactly half the spins land on black?

Answer 1

The probability of exactly half the spins landing on black can be calculated using the binomial probability formula.

Step-by-step explanation:

The probability of exactly half the spins landing on black can be calculated using the binomial probability formula. Let's assume there are a total of n spins, where n is an even number. The probability of a single spin landing on black is 50% or 0.5. To find the probability of exactly half the spins landing on black, we need to find the probability of achieving exactly n/2 black spins out of n spins. This can be calculated using the formula:

P(k) = C(n, k) * (0.5)^k * (0.5)^(n-k)

Where P(k) is the probability of getting k black spins, C(n, k) is the number of ways to choose k black spins out of n spins (also known as combination), and (0.5)^k and (0.5)^(n-k) represent the probabilities of getting k black spins and (n-k) white spins respectively.

Let's consider an example where n = 4. The possible outcomes are:

BBBB, BBWW, BWWB, WWBB, WBWB, WBWB, WWBW, WBBW

Out of these, there are 4 outcomes where exactly half the spins land on black (BBBB, BBWW, WWBB, WBWB). So, the probability of exactly half the spins landing on black when n = 4 is 4/8 or 0.5.

Answer 2

The probability of exactly half the spins landing on black is determined by using the binomial probability formula, considering that each spin has a 50% chance to land on black, which is similar to the probability of a coin toss resulting in heads.

Step-by-step explanation:

The student is asking about the probability of a specific outcome when spinning a wheel or flipping a coin multiple times. Specifically, the question is what the probability is that exactly half the spins land on black. To calculate such probabilities, we use rules from combinatorics and probability theory. For instance, if there are a total of 10 spins and we want exactly 5 black spins, we need to determine the number of ways to choose 5 out of 10 (which is a binomial coefficient) and then multiply it by the probability of each combination occurring. The probability of landing on black in a single spin is assumed to be 50%, similar to the chance of a coin landing on heads in a coin toss.

Using the formula for the binomial probability P(X=k) = (n choose k) * p^k * (1-p)^(n-k) where p is the probability of one successful outcome (landing on black), k is the number of successful outcomes we want, and n is the total number of trials, we can calculate the probability of exactly half the spins landing on black. If n is even, the calculation is straightforward, but if n is odd, it's impossible to have exactly half the outcomes.