If there is a 2.3% chance of something happening, what are the odds it NOT happening 400 and 500 times in a row?

Question

See ref: https://www.wyzant.com/resources/answers/803108/if-there-is-a-2-3-chance-of-something-happening-what-are-the-odds-it-not-ha

A gentlemen helped me and I thank him. However I did his calculation and I come up with answers that have one less 0 than him :

''chance of not happening = 100%-2.3% = 97.7% on each time

for 400 times the chance is (.977)^400

plug that into a scientific calculator to get

9.07455814 x 10^-5 which is

0.00000907455814 or

0.000907455814% chance of never happening in 400 tries''

I did the calculations on a scientific calculator : (.977)^400 and 9.07455814 x (10^-5) And they both gave 0.0000907455814 (0.00907455814%), which is one less 0 than the answer provided. Given I am about to calculate quite a few of theses, I was just wondering if it was just a typo or if I missed a notion.

Is the right answer

0.00000907455814

or

0.0000907455814

Thank you again in advance.

Answer 1

4 zeroes before the 9 is correct

5 is wrong

just a few observations about your answers, but I get into the zeroes towards the end:

.use the "rule of 9's"

add the digits

9+7+7= 23, 2+3= 5

5x5 =25 2+5=7

25x25 = 625 6+2+5 =13 1+3= 4

625x625 = 390625, a number that has 7 as the result

390625^2 results in 6

390625^4 results in 9

390625^8 results in 9

.977^400 results in 9

your answers is a long decimal that ends in 4

.977^400

is correct. = .00009....

.977^2 = .954529

.977^4 = .9542529^2 = .911125611

.977^8 = 0.83014988

.977^16 = 0.689148824

this calculator result seem to drop some decimals

...the last digit of your final calculation ends in a decimal digit 9

.977 to any power will go through a cycle of 7,9,3,1, then repeats

7^1 =7, 7^2 = 49 7^3 = 343, 7^4 = 2401. the last digit each time follows a repeating pattern, 7,9,3,1

7^8 ends in 1

7^16 ends in 1

7^400 ends in 1

Your calculation ends in 4, although best guess is it's truncated, leaving off other decimals

.977^128 = .050874968 = about 5 x 10^-2

.977^256= .002588262452 = about 2.59 x 10^-3

.977^384 = about 1.32 x 10^-4

.977^400 = .977^384(.977^16) = about 1.32 x 10^-4 x .69 = .908 x 10^-4

= 9.08 x 10^-5

you should have 9 x 10^-6 if you got 5 zeroes to the right of the decimal point,but 9x10^-5 and 4 zeroes is correct.

.01 = 1 x 10^-2 the absolute value of the exponent is 1 greater than the number of zeroes.