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Odds of rolling a ten sided die 200,000 times and getting the number 4 only 8 ti...

Jan 29, 2024

Odds of rolling a ten sided die 200,000 times and getting the number 4 only 8 times?

Solution by Steps

step 1

To find the probability of rolling a ten-sided die 200,000 times and getting the number 4 exactly 8 times, we use the binomial probability formula

step 2

The binomial probability formula is given by

P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}

, where

n

is the number of trials,

k

is the number of successes,

p

is the probability of success on a single trial, and

\binom{n}{k}

is the binomial coefficient

step 3

In this case,

n = 200,000

k = 8

, and

p = \frac{1}{10}

because there is a 1 in 10 chance of rolling a 4 on a ten-sided die

step 4

The binomial coefficient

\binom{n}{k}

can be calculated using the formula

\binom{n}{k} = \frac{n!}{k!(n-k)!}

step 5

Plugging the values into the binomial probability formula, we get

P(X=8) = \binom{200,000}{8} \left(\frac{1}{10}\right)^8 \left(1-\frac{1}{10}\right)^{200,000-8}

step 6

Calculating this probability directly is computationally intensive due to the large numbers involved, so approximation methods or software tools are typically used to find the probability

Answer

The exact probability is given by the formula in step 5, but it requires computational tools to calculate.

Key Concept

Binomial Probability

Explanation

The key concept is the binomial probability, which is used to determine the likelihood of a given number of successes in a fixed number of independent trials, each with the same probability of success.