Question

Same topic:

1. Consider all passwords of length 10 made up of numbers, lower-case letters, and upper-case letters (from the English alphabet).

a) If you pick such a password at random, what is the likelihood that it contains exactly 5 lower-case letters?

b) If you pick such a password at random, what is the likelihood that it contains at most 9 lower-case letters?

2. Suppose you’re an ornithologist who specializes in American crows (Corvus brachyrhynchos), and you’ve collected data on 100 such birds. Each bird was classified by sex (male or female), and by size (large or small).

The counts follow:

• Male, Large: 38,

• Male, Small: 13,

• Female, Large: 23,

• Female, Small: 26.

Based on these numbers, what is the probability that a randomly chosed crow from this group is large?

What is the probability that a crow is large if it is already known to be male? Is the event of being large independent of the event of being male?

Answer 1

1.

Number of lower case letters = 26

Number of upper case letters = 26

Number of numbers = 10 ( 0 to 9 )

Total options for a digit = 26+26+10 = 62

Total number of 10 digit passwords we can generate using all the above 3 = .

(a)

Number of passwords containing exactly 5 lower case letters =

( as we use any 5 out of 26 lower case letters and 5 out of 36 others and repetition is allowed )

liklihood is is given by probability :

probability of a randomly chosen password being a password of exactly 5 lower case letters is

(b)

Number of passwords containing at most 9 lower case letters

= Total number of passwords - Number of passwords with exactly 10 lower case letters

This time the probability is

2.

Number of crows = 100	Male	Female
Large	38	23
Small	13	26

a) total number of large crows = 38 + 23 = 61

The probability of randomly chosen crow is a large one

b) total number of male = 38 +13 = 51 ( let denote the corresponding event by M and N corresponds to large crows )

total number of large male crow = 38 ( This is then

The probability that  that a crow is large if it is already known to be male

Now probability that a crow is male

probability that a crow is large

Product of both probabilities

Now the probability of large male crow

So, the events of being large and being a male are not independent.