Justify whether or not the following is a binomial.
Rolling a fair die 3 times and observing the number of times a 6 is thrown.
I know that the binomial rules are :
1. The number of trials in the experiment is fixed.
2. There are two outcomes of each trial: "success" and "failure".
3. The probability of success in each trial is the same.
4. The trials are independent of each other.
1. Fixed - are 3 trials.
2. There is a success if it lands on 6 and failure if it lands on any number that is not 6.
success/p =1/6 fail/q=5/6
3. 1/6+5/6=1
4. I don't know how to put together the words to explain this.
For 4, if they're independent, it would look something like this right?
E= { SFF, SFS, SSF, FFS, FSS, FSF, FFF}
1 s 2s 2s 1s 2s 1s 0s
With s standing for success and the numbers being the number of successes.
(1/6)(1/6)(1/6) = 1/216
Independent means that the outcome of the first trial does not impact the second trial.
In this case good. yes independent
BUT if you had 6 cards numbered 1 to 6
and you drew the first and had the number 3
the chance was 1/6 of getting three.
HOWEVER if you do NOT put that BACK, the chance of getting three will be zero on the next draw and the chance of getting the number 2 will be 1/FIVE. Second draw therefore DEPENDENT on first draw..
To determine whether the given scenario can be classified as a binomial, we need to examine the four fundamental rules of a binomial experiment:
1. Fixed Number of Trials: The scenario involves rolling a fair die three times. This satisfies the requirement of a fixed number of trials.
2. Two Outcomes: In each trial, there are two possible outcomes - success or failure. Success occurs when a 6 is rolled, while failure occurs when any other number (1, 2, 3, 4, or 5) is rolled.
3. Constant Probability: The probability of success remains the same for each trial. In this case, the probability of rolling a 6 is 1/6, while the probability of rolling any other number is 5/6. Therefore, the condition of constant probability is met.
4. Independence: The trials must be independent of one another, meaning the outcome of one trial does not affect the outcome of the others. Rolling a die is an independent event since the result of one roll does not influence the result of subsequent rolls. Therefore, the independence condition is satisfied.
Based on our analysis of the four rules, we can conclude that the given scenario of rolling a fair die three times and observing the number of times a 6 is thrown meets all the criteria to be classified as a binomial experiment.