so 3 times a probability of 1 in 3, with a table it gives 19/27 so +- 70 %, i would like to know how to do it with a formula
Responses (1)
3
∑((3Cx) (1/3)^x (2/3)^(3-x))
x=1
This formula solves for things happening a ranged amount of times. This formula is a summation, meaning that the variable below the sigma (the character that looks like an "E"), in this case "x," is inputted into the formula to the right of the sigma for every integer between the minimum below the sigma (1) and the maximum above the sigma (3). Then the answers from each input are added together to form your answer.
The leftmost segment of the formula to the right of the sigma (3Cx) is known as a combination. A combination determines the total number of ways something can be grouped. The number to the left of the "C" in this formula is the number of attempts and to the right is the number of successes. It is an "x" because what we are solving for is the probability of a success 1, 2, or 3 times, which is handled by the summation. Your calculator might be missing the combination function, so you can use (3!/(x!(3-x)!)) in place of (3Cx), which is a combination written out. The "!" stands for a factorial. A factorial multiplies the number preceding it by every counting number less than it; for example: 3!=3*2*1=6.
The middle segment of the formula to the right of the sigma has a fraction that represents the chance of success in one attempt. It is raised to the power of the number of successes calculating for.
The last segment of the formula to the right of the sigma has a fraction that represents the chance of failure in one attempt. It is raised to the power of the number of failures calculating for, thus the number of attempts minus successes.
The formula does give you your expected answer of approximately 70%. If you were already aware of the concepts I explained, then sorry for making my response so wordy. I just wanted to ensure full understanding of my response.