Binomial Distribution

#1
One classical experiment on ESP (extrasensory perception) tests for the ability of an individual to show telepathy-that is, to read the mind of another individual. This test uses five cards with different designs, all known to both participants. In a trial, the "sender" sees a randomly chosen card and concentrates on its design. The "receiver" attempts to guess the identity of the card. Each of the five cards is equally likely to be chosen, and only one card is the correct answer at any point.

a) Out of 10 trials, a receiver got four cards correct. What is her success rate? What is her expected rate of success, assuming she is only guessing?

For the success rate, is it 4/10=.4? As for the expected rate of success, I am not sure I understand what it is asking for.

b) Is her higher actual success rate reliable evidence that the receiver has telepathic abilities? Carry out the appropriate hypothesis test.

Which hypothesis test would I use for this?
 
#2
a)

What is her success rate?

40%.

What is the expected success rate?

Consider 1 "trial" to be the random variable \(X_i\). Let \(X_i\) equal 1 if she guesses the correct card and 0 if she guesses incorrectly. Then with 5 cards, \(X_i\) = 1 has probability 0.2, and \(X_i\) = 0 has probability 0.8. From this we know that \(E[X_i] = 0.2\).

For 10 trials, we have E[\sum_{i=1}^10 X_i] = \sum_{i=1}^10 E[X_i] = 10*0.2 = 2.

b)

Dunno.
 
#3
a)


What is the expected success rate?

Consider 1 "trial" to be the random variable \(X_i\). Let \(X_i\) equal 1 if she guesses the correct card and 0 if she guesses incorrectly. Then with 5 cards, \(X_i\) = 1 has probability 0.2, and \(X_i\) = 0 has probability 0.8. From this we know that \(E[X_i] = 0.2\).

For 10 trials, we have E[\sum_{i=1}^10 X_i] = \sum_{i=1}^10 E[X_i] = 10*0.2 = 2.



The estimated success rate cannot be greater than 1 since it is a percentage.

Her estimated success rate is just 0.2 or 20%. This is because:
The probability of guessing the correct answer for one trial is 1/5=0.2 or 20%
Each trial is independent from the other trials which means that if she guesses the correct answer on one trial, she still only has 20% chance of guessing the correct answer on any subsequent trial. So since each trial has a 20% success rate, the total overall expected success rate is also 20%.
 

Outlier

TS Contributor
#4
Mr. Bean seems to have ex·oph·thal·mos.

b) Is her higher actual success rate reliable evidence that the receiver has telepathic abilities? Carry out the appropriate hypothesis test.
On this site someone posted a 40 min. YouTube link on this subject in response to a question I had. One guy did have ESP to a high level of significance but the test givers did not believe their own evidence.

The subject had to draw a complex picture that was in another room. The only thing he left out was a shed in the foreground.

I once asked a woman to tell me what single digit I was thinking of.
She got the first two wrong, but the third one was right, including what colors the digit had in my mind.
At that, she announced she was scared and did not want to continue.