A fair die is rolled 4 times
a) create a probability distribution for the number of times the die rolls a 3.
Do I solve this using the bernoulli trials formula? Or should I just examine the frequencies like this?
E= Rolling a 3 A= Not a 3
EAAA 1 3
EEAA 2 3s
EEEA 3 3s
EEEE 4 3s
AAAA 0 3s, etc......
To create a probability distribution for the number of times the die rolls a 3, you can use the approach of examining the frequencies as you have described. Here's how you can determine the probabilities:
1) Start by listing all the possible outcomes when rolling the die 4 times. Each outcome is a sequence of four letters representing whether it rolled a 3 (E for "event" or "success") or not (A for "absence" or "failure").
The possible outcomes are:
EEEE
EEEA
EEAE
...
AAAA
2) Count the number of times the die rolls a 3 in each outcome. From your example, you have already provided the counts for each outcome.
3) To find the probability of each outcome, divide the count of the desired event (rolling a 3) by the total number of outcomes possible (which is 6^4, or 1296).
For example:
P(4 3s) = (number of outcomes with 4 3s) / (total number of outcomes) = 1 / 1296
P(3 3s) = (number of outcomes with 3 3s) / (total number of outcomes) = 4 / 1296
P(2 3s) = (number of outcomes with 2 3s) / (total number of outcomes) = 6 / 1296
4) Repeat this process for each possible number of times the die rolls a 3.
In the end, you will have a probability distribution that shows the probabilities for each possible outcome (number of times a 3 occurs).