# S1 Probability - Binomial & Geometric Distribution

• AntSC
In summary, the conversation discusses a board game where four players take turns throwing two fair dice. A double roll moves the player forward six squares, otherwise they move one square. The conversation then outlines how to find the probability of a double occurring on the third throw, exactly one player getting a double in the first round, and a double occurring once in four of the first five rounds. The solution involves using geometric and binomial problems, with the final probability being 0.068.
AntSC

## Homework Statement

Four players play a board game which requires them to take it in turns to throw two fair dice. Each player throws the two dice once in each round. When a double is thrown the player moves forward six squares. Otherwise the player moves forward one square

## Homework Equations

Find:
a) the probability that the first double occurs on the third throw of the game
b) the probability that exactly one of the four players obtains a double in the first round
c) the probability that a double occurs exactly once in 4 of the first 5 rounds

## The Attempt at a Solution

a) A geometric problem.
$\mathbb{P}\left ( X=3 \right )=\frac{1}{6}\left ( \frac{5}{6} \right )^{2}=0.1157$

b) A binomial problem.
$\mathbb{P}\left ( X=1 \right )=\binom{4}{1}\frac{1}{6}\left ( \frac{5}{6} \right )^{3}=0.3858$

I've checked the answers for part a) and b) and they're correct.

c) This part i have no idea. I've wracked my brains for ages and i can't get clear about making sense of 'once in 4
of the first 5 rounds'. Any help would be brilliant

I think it means that in four of the first five rounds a double happens exactly once. Could be any four of the five. Not sure what should be assumed for the other round... any result at all? No doubles? Any number of doubles except one?

Isn't saying 'once in four of the first five', the same as saying 'once in the first five'?

AntSC said:
Isn't saying 'once in four of the first five', the same as saying 'once in the first five'?
Not if it means "once in each of some four of the first five".

So i take the probability of once in four and then multiply it by how many ways i can get 4 rounds from a total of 5 rounds, namely $\binom{5}{4}$ ?

Or $\binom{5}{4}\mathbb{P}\left ( X=1 \right )$ ?
This is clearly wrong. But it's the closest to making sense to me right now.

AntSC said:
So i take the probability of once in four and then multiply it by how many ways i can get 4 rounds from a total of 5 rounds, namely $\binom{5}{4}$ ?
No, you need it to happen in exactly four of the five rounds.

I get it now.
It's a new binomial problem with parameters $X\sim B\left ( 5, 0.3858 \right )$
So $\mathbb{P}\left ( X=4 \right )=\binom{5}{4}p^{4}q=0.068$
Checked the answer, and it agrees.
Thanks for the help

## 1. What is the difference between binomial and geometric distribution?

Binomial distribution is used to calculate the probability of a specific number of successes in a fixed number of independent trials with a constant probability of success. Geometric distribution is used to calculate the probability of the number of trials needed to achieve the first success in a series of independent trials with a constant probability of success.

## 2. How do you calculate the mean and standard deviation for a binomial distribution?

The mean for a binomial distribution is calculated by multiplying the number of trials by the probability of success. The standard deviation is calculated using the formula √(n * p * (1-p)), where n is the number of trials and p is the probability of success.

## 3. Can binomial distribution be used for non-binary events?

Yes, binomial distribution can be used for events with more than two possible outcomes as long as the probability of success remains constant for each trial.

## 4. What is the difference between a success and a trial in binomial distribution?

A success in binomial distribution refers to the desired outcome, while a trial refers to a single attempt or occurrence of the event. For example, if you are flipping a coin three times to see how many times you get heads (success), then each flip would be considered a trial.

## 5. What is the formula for calculating the probability of a specific number of successes in a binomial distribution?

The formula for calculating the probability of x successes in a binomial distribution is P(x) = (nCx) * (p^x) * ((1-p)^(n-x)), where n is the number of trials, p is the probability of success, and x is the number of desired successes.

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