I need help with this pre-calculus problem?

Question

Experience has shown that 14% of purchases are returned to a store. Find the probability that either 2 or 3 of the next 6 will be returned.

Can you show me how to do this?

Answer 1

Probabilty of 2 of 6 purchases are returned:

Long form: Write out the probability for 1 permutation that achieves the result. Then, multiply that answer the # of possible permutations:
.14 * .14 * .86 (because .86 = 1 - .14)* .86 * .86 * .86 = .14^2 * .86^4 = 0.010721359936

To show all permutatations (R for return, N for not returned):
RRNNNN
RNRNNN
RNNRNN
RNNNRN
RNNNNR
NRRNNN
NRNRNN
NRNNRN
NRNNNR
NNRRNN
NNRNRN
NNRNNR
NNNRRN
NNNRNR
NNNNRR
This is 15 permutations, which can be simplified as a summation of (# of purchase - # of returns + 1). Summation of 5 = 5 + 4 + 3 + 2 + 1 = (n * n + 1) / 2 = 15.
15 * .14^2 * .86^4 = 0.16082039904 = 16.1% chance for 2 returns.

3 returns - done in short form now that we've seen how the long form plays out:
.14^3 (3 returns) * .86^3 (3 kept) * (summation of (6 - 3 + 1) = summation of 4 = (4 * 5) /2 = 10)
.14^3 * .86^3 * 10 = 1.75% chance for 3 returns