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Here is **Question 4** from your assignment:

### **Question 4 [5 + 6 + 3 = 14 marks]**

An extrasensory perception (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.

**(a)** How many possibilities are there for the order in which the subject writes down the numbers?

**(b)** 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?

**(c)** Based on your result from part (b), what would you conclude if the subject writes down the numbers in the correct order? Explain your answer.

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### **Solutions:**

**(a)** The subject can write down any permutation of the 10 numbers (1 to 10), so the total number of possible orders in which the subject can write the numbers is simply the number of permutations of 10 distinct objects. This is calculated as \( 10! \).

\[
10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800
\]

So, the number of possibilities for the order in which the subject writes down the numbers is **3,628,800**.

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**(b)** If the subject has no ESP and is guessing randomly, the probability that they guess the exact order in which the cards are arranged is simply the probability of picking one specific permutation out of the \( 10! \) possible permutations.

\[
\text{Probability} = \frac{1}{10!} = \frac{1}{3,628,800}
\]

So, the probability that the subject writes the numbers in the correct order is \( \frac{1}{3,628,800} \), which is approximately \( 2.755 \times 10^{-7} \).

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**(c)** If the subject writes down the numbers in the correct order, given the extremely low probability calculated in part (b), it would suggest that the subject either has some form of ESP or that some other external factor (such as cheating or a flaw in the experiment) influenced the result. Since the probability of guessing the correct order is extremely small, it would be highly unlikely that the result happened purely by chance.

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Would you like further explanation on any part of this? Here are some related questions that might deepen your understanding:

1. How do you compute factorials for different numbers?
2. What are permutations and how do they differ from combinations?
3. Can you explain more about probability theory and why small probabilities suggest unlikely outcomes?
4. How might biases or experimental design affect the outcomes in psychology experiments?
5. How does probability theory apply to real-world decision-making?

**Tip**: Always double-check factorial calculations, especially for large numbers like \(10!\), since any small mistake can greatly affect the final result.

Explore the concept of permutations and probability in an extrasensory perception (ESP) experiment. Learn how to calculate the number of possible outcomes and the probability of correctly guessing the order of shuffled cards in this detailed mathematical analysis.

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