Let the probability of winning a bet in a roulette

bobbie71G

bobbie71G

Answered question

2021-09-20

Let the probability of winning a bet in a roulette game is 1/38.Find the probability of winning at least 1 time in 38 games with the Binomial formula and the Poisson's approximation to the Binomial distribution.

Answer & Explanation

Leonard Stokes

Leonard Stokes

Skilled2021-09-21Added 98 answers

Step 1
The binomial probability formula is used when it satisfies the following conditions:
1. There are only two possible outcomes.
2. The number of trials is fixed.
3. Probability of success in each trial is equal.
4. Each trial is independent.
The formula to find the probability of binomial random variable is
P(x)=n!x!(nx)!px(1p)x, where n is the number of trials, x is the number of successful trials, and p is the probability of success in each trial.
Poisson's approximation is used, when there are a large number of trials and the probability of success is very less. The formula to find the probability using Poisson's approximation is P(x)=eλλxx!, where λ=np.
Step 2
The probability of success is p=138, and the total number of trials is n=38. Now, it is asked to find the probability that there is at least 1 successful trial, which means x1. This probability will be complement to P(x=0) and thus P(x1)=1P(x=0).
The value of P(x=0) can be found by substituting p=138 and n=38 in the formula of binomial probability distribution.
P(x1)=1P(0)
=138!0!(380)!(138)0(1138)380
=111(3738)38
=10.363
0.637
Step 3
Similarly, find the value of P(x=0) using the formula of Poisson's approximation P(x)=eλλxx! and first find the value of λ.
λ=38(138)
=1
Now, use the value λ=1 and x=0 in the formula P(x)=eλλxx! and find the value of P(x=0) and compute the probability of at least 1 success.
P(x1)=1P(0)
=1e1(1)00!
=1e1
=10.368
0.632
The probability computed from both the formulas is approximately equal.

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