Hello
What's the probability of correctly guessing three out of five cards numbered one to nine?

thx

Give some details. Do you have to guess rank and suit or just the rank? How many cards are in the deck? Maybe 40, with 10 ranks and four suits?

if there are 9 cards and someone chooses 5 at random and asks you to guess, the probability of guessing EXACTLY 3 of them right is 5C3*4C2/9C5 = 10*6/126 = 0.4762

all probabilities are:
all 5: 1/126
4 0.1587
3: 0.4762
2: 0.3175
1: 0.0397

Actually this test is just about the numbers. Not about playing cards.
So let's say I have to guess the rank only (one to nine). Suit doesn't matter. Each card can be numbered 1-9
There are 5 of them. I guess three correctly.

I am asking about the test in the video below. I think the neuroscientist is calculating the probability incorrectly at the 4:10 mark.
She calculated 1/9*1/9*1/9 (for three good guesses out of five cards) :

https://www.youtube.com/watch?v=N7dDMfSFfLY

if this game is played in a sequence without replacing the cards, then the probabilities would be like this (g means good guess, b bad):
bbggg : 8/9 * 7/8 * 1/7 * 1/6 * 1/5
bgbgg : 8/9 * 1/8 * 6/7 * 1/6 * 1/5
bggbg : 8/9 * 1/8 * 1/7 * 5/6 * 1/5
bgggb : 8/9 * 1/8 * 1/7 * 1/6 * 4/5
gbbgg : ...
gbgbg
gbggb
ggbbg
ggbgb
gggbb

total probability should be 0.0235 or 2.35%

and if she's replacing the cards that are drawn out, then it's (1/9)^3 * (8/9)^2 * 5C3 = 0.0108 = 1.08%

There is nothing in this that suggests that there can't be repeat numbers. If each card can be randomly selected from 1 to 9, then the 1/9 * 1/9 * 1/9 would be correct - IF you only had 3 tries. Since you actually get 5 attempts, her calculation is incorrect.

The way I understand her in the video would be "get 3 out of 5 correct - or better" (since we're talking about a 'psychic genius' a 'better' score obviously is also a 'psychic genius')

The probability of getting 5 correct is:
(1/9)^5

The probability of getting exactly 4 correct is:
(1/9)^4*(8/9)* 5!/(4!*1!)

The probability of getting exactly 3 correct is:
(1/9)^3*(8/9)^2 * 5!(3!*2!)

...add that all up and I get roughly 1%

Read: if you drive around and test a few kids every day you'll end up with no shortage of 'psychically gifted' kids that really aren't.

For comparison..just taking her 1/9 * 1/9 * 1/9 approach is off by an order of magnitude. That would peg the probability at just 0.1%