Problem 15BSC

Deciphering Messages The Central Intelligence Agency has specialists who analyze the frequencies of letters of the alphabet in an attempt to decipher intercepted messages that are sent as ciphered text. In standard English text, the letter r is used at a rate of 6%.

a. Find the mean and standard deviation for the number of times the letter r will be found on a typical page of 2600 characters.

b. In an intercepted ciphered message sent to Iran, a page of 2600 characters is found to have the letter r occurring 178 times. Is this unusually low or high?

Answer :

Step 1 of 1

a)

On a typical of n=2600 characters, the probability of success p=6%.

Then p=6/100 = 0.06 because the letter is used at a rate of 6%.

This gives the probability of failure, q = 1 - p.

q = 1 - p = 1 - 0.06 =0.94

Therefore the mean and the standard deviation for the given binomial distribution is given as

The Mean of the formula is

μ=np

μ =(2600) (0.06)

μ = 156

Therefore the mean μ =156

The formula of the standard deviation is given by

σ =

σ =

σ =

σ =

σ = 12.1095

Therefore the standard deviation is σ = 12.1095

b)

We know that μ =156 and σ = 12.1095

Maximum usual values μ + 2σ and Minimum usual values μ-2σ

= ( μ ± 2σ)

= (156 ± 2(12.1905) )

= (156 ± 24.381 )

= (156+24.381 , 156-24.381 )

= (180.381 , 131.619 )

Any value in between 180.381 and 131.619 is not considered unusually since it lies with 2 standard deviation from the mean.

Yes.In an intercepted ciphered message sent to Iran, a page of 2600 characters is found to have the letter r occurring 178 times.

A value of 178 could not be considered unusually.