An ESP experiment is conducted by a psychologist. For part of the experiment, the psychologist takes 10 cards numbered 1-10 and shuffles them. Then she looks at the cards one at a time. While she looks at each card, the subject writes down the number he thinks is on the card.

If the subject has no ESP and is just guessing each time, what is the probability that he writes down the numbers in the correct order, that is, in the order that the cards are actually arranged?

The answer is 2.67E-7.

To solve this problem, we need to calculate the probability of guessing the correct order of the shuffled cards, assuming the subject has no ESP and is purely guessing.

Since there are 10 cards numbered from 1 to 10, there are 10! (10 factorial) possible ways the cards can be arranged. This is because for the first card, there are 10 possibilities, for the second card, there are 9 possibilities remaining, for the third card, there are 8 possibilities remaining, and so on.

Thus, the total number of possible arrangements is 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3,628,800.

Given that the subject has no ESP and is guessing randomly, each card has an equal chance of being assigned any number from 1 to 10. Therefore, for the first card, the probability of guessing the correct number is 1/10, for the second card it is 1/10, for the third card it is 1/10, and so on.

Since the subject has to guess the correct order of all 10 cards, we multiply the individual probabilities together:

P(guessing correct order) = (1/10) x (1/10) x ... x (1/10) (10 times)

This simplifies to (1/10)^10 = 1/10^10.

Calculating this probability gives us:

1/10^10 = 1/10,000,000,000 = 0.0000000001

In scientific notation, this is 1.0 x 10^-10.

Therefore, the probability of guessing the correct order with no ESP is 1.0 x 10^-10 or 1 in 10 billion.

So the answer you provided, 2.67E-7, appears to be incorrect. The correct probability is significantly smaller.