Question 476720: In a group of 50 men, 16 are bald, 20 have beards, and 4 are bald and have beards. A man is selected at random. (a) find the probability he is bald given he has a beard. (b) find the probability he has a beard given he is bald.
Found 2 solutions by jorel1380, Edwin McCravy:
Answer by jorel1380(3719)   (Show Source):
Answer by Edwin McCravy(20060)  (Show Source):
You can put this solution on YOUR website! In a group of 50 men, 16 are bald, 20 have beards, and 4 are bald and have beards. A man is selected at random. (a) find the probability he is bald given he has a beard. (b) find the probability he has a beard given he is bald.
The other tutor got the correct answers in decimal form,
but didn't give the slightest clue as to how he got them.
Make this chart, or an equivalent Venn diagram:
| Bald | Non-Bald | Totals
Beard | | |
----------------------------------------------
Beardless| | |
----------------------------------------------
Totals | | |
We can fill in the 50 for the total men in the bottom
right, 16 for the total bald, 20 for total with
beards and 4 bald and bearded, straight from the
given numbers, like this:
| Bald | Non-Bald | Totals
Bearded | 4 | | 20
----------------------------------------------
Beardless| | |
----------------------------------------------
Totals | 16 | | 50
Next by subtraction we get that
1. Left column: there are 16-4 or 12 beardless bald men.
2. Top row: there are 20-4 = 16 bearded non-bald.
3. Bottom row: there are 50-16 = 34 total non-bald men.
4. Right column: there are 50-20 = 30 total beardless men
| Bald | Non-Bald | Totals
Bearded | 4 | 16 | 20
----------------------------------------------
Beardless| 12 | | 30
----------------------------------------------
Totals | 16 | 34 | 50
Finally we can fill in the number of beardless
non-bald men either of two ways either by
1. Middle column: 34-16 = 18
or
2. Middle row: 30-12 = 18.
| Bald | Non-Bald | Totals
Bearded | 4 | 16 | 20
----------------------------------------------
Beardless| 12 | 18 | 30
----------------------------------------------
Totals | 16 | 34 | 50
Now we can choose either to do the conditional
probability problems by:
1. Reduced sample space mthod
2. By the formula method
Let's do it first by the reduced sample space method.
(a) find the probability he is bald given he has a beard.
Since we are given that he has a beard. we can eliminate
everything except the top row, so we just look only at
this part of the above chart:
| Bald | Non-Bald | Totals
Bearded | 4 | 16 | 20
----------------------------------------------
Using this reduced sample space we see that the
desired conditional probability
P(bald|bearded) = 4/20 = 1/5
==================================================
(b) find the probability he has a beard given he is bald.
Since we are given that he is bald, we can eliminate
everything except the left column, so we just look at
this part of the chart above:
| Bald |
Bearded | 4 |
---------------------
Beardless| 12 |
---------------------
Totals | 16 |
Using this reduced sample space we see that the
desired conditional probability
P(bearded|bald) = 4/16 = 1/4
===================================================
P(X and Y)
We can also do these by this formula: P(X|Y) = ————————————
P(Y)
From the entire chart:
| Bald | Non-Bald | Totals
Bearded | 4 | 16 | 20
----------------------------------------------
Beardless| 12 | 18 | 30
----------------------------------------------
Totals | 16 | 34 | 50
(a)
P(bald and bearded) 4/50 4 1
P(bald|bearded) = ——————————————————— = —————— = ———— = ———
P(bearded) 20/50 20 5
(b)
P(bearded and bald) 4/50 4 1
P(bearded|bald) = ——————————————————— = —————— = ———— = ———
P(bald) 16/50 16 4
Either method for conditional probability is correct.
Edwin
|
|
|
