7.6: Probability

If a coin is tossed three times, what is the probability of getting exactly two heads? at least two heads? no heads?

Probability is defined as the number of ways the event in question can happen divided by the number of total possibilities. We can define the collection of all possible outcomes in an experiement with all outcomes equally likely as the sample space of the experiment. Then the number of ways the event in question can happen can be defined as \(n(E)\) and size of the sample space can be defined as \(n(S) .\) In this notation, the probability of an event \(P(E)\) is defined below:
\[
P(E)=\frac{n(E)}{n(S)}=\frac{\text { number of elements in } E}{\text { number of elements in } S}
\]
The question at the opening of this section addresses a situation in which a coin is tossed three times. We can use a tree diagram to examine the sample space of this situation.
The possibilities in this experiment are:
\(\{\mathrm{HHH}, \mathrm{HHT}, \mathrm{HTH}, \mathrm{HTT}, \mathrm{THH}, \mathrm{THT}, \mathrm{TTH}, \mathrm{TTT}\}\)
since there are eight possibilities, with 3 of them having exactly two heads, the probability is \(\frac{3}{8}=0.375\)

The probability of getting at least two heads is \(\frac{4}{8}=\frac{1}{2}\) or 0.5
The probability of getting no heads is \(\frac{1}{8}=0.125\)
Example
A five-card poker hand is drawn from a standard deck of 52 cards. What is the probability that all five cards are hearts?

The number of ways to draw five cards from a deck of 52 is \(_{52} C_{5}\). This is a combination since the order that the cards are drawn does not affect the outcome.
The number of ways of choosing 5 hearts from the 13 in the deck is \(13 \mathrm{C}_{5}\)
So the probability of drawing five hearts from a deck of 52 cards is \(\frac{13 C_{5}}{52 C_{5}}=\) \(\frac{1287}{2598960} \approx 0.000495\)
Example
A bag contains 20 tennis balls, of which four are defective. If two balls are drawn at random from the bag, what is the probability that both are defective?

The sample space is the number of ways to choose two balls from the bag: \(\quad_{20} C_{2}\)
since there are 4 defective balls, the number of ways to draw two of them is \(_{4} C_{2}\)
So, the probability that both balls will be defective is \(\frac{4}{20 C_{2}}=\frac{6}{190} \approx 0.0316\)
We can also calculate the probability of drawing one defective ball:
\(\frac{_{4} C_{1} *_{16} C_{1}}{_{20} C_{2}}=\frac{64}{190} \approx 0.3368\)

And the probability of drawing no defective balls:
\(\frac{_{16} C_{2}}{_{20} C_{2}}=\frac{120}{190} \approx 0.6316\)
Notice that \(6+64+120=190\) and that \(0.0316+0.3368+0.6316=1\) That is to say that the probabilities for all the possible events should add up to 1

This allows us to compute probabilities based on the probability that an event won't happen.

If the probability of an event is \(P(E),\) then the probability that the event will not happen is \(1-P(E)\)

Example
An urn contains 10 red balls and 15 green balls. If six of the balls are drawn at random, what is the probability that at least one of them is red?

We can calculate this probability by finding the probability that no red balls are drawn. The number of ways to draw six green balls is \(_{15} C_{6} .\) The number of ways to draw six balls from the urn is \(_{25} C_{6}\). So the probability of drawing six green balls is:
\(\frac{_{15} C_{6}}{_{25} C_{6}}=\frac{5005}{177,100} \approx 0.02826\)
This means that the probability of drawing at least one red ball is \(1-0.02826=\) 0.97174
We could also find this answer by adding up the possibilities of drawing one red ball, two red balls and so on, up to six red balls:
\(\frac{_{10} C_{1} *_{15} C_{5}}{_{25} C_{6}}+\frac{10 C_{2} *_{15} C_{4}}{25 C_{6}}+\frac{10 C_{3} *_{15} C_{3}}{25 C_{6}}+\frac{10 C_{4} *_{15} C_{2}}{25 C_{6}}+\frac{10 C_{5} *_{15} C_{1}}{25 C_{6}}+\frac{10 C_{6}}{25 C_{6}}\)
\(=0.1696+0.3468+0.3083+0.1245+0.0213+0.0012=\)
0.9717

Example
Given 100 computer components, it is known that 10 of the 100 are defective. If someone were to choose 6 of these components at random, what is the probability that:
a) two of them are defective?
b) at least 1 of them is defective?
There are \(_{10} C_{2}\) ways to choose 2 out of the 10 defective components and \(_{90} C_{4}\) ways to choose 4 non-defective components. The size of the sample space (the number of ways to choose 6 components from the 100 ) is \(_{100} C_{6}\)
So, the probability for part (a) would be:
\(\frac{_{10} C_{2} *_{90} C_{4}}{_{100} C_{6}} \approx 0.096\)
The probability in part (b) is most easily computed by finding the probability that no defective components are selected and then subtracting that value from 1
\(1-\frac{_{90} C_{6}}{_{100} C_{6}} \approx 1-0.5223 \approx 0.4777\)

Exercises 7.6
SET I
1)
\(\quad\) a) If a coin is tossed two times, describe the sample space.
\(\quad\) b) Find the probability of getting exactly two heads.
\(\quad\) c) Find the probability of getting at least one head.
\(\quad\) d) Find the probability of getting exactly one head.
2)
\(\quad\) a) If a coin is tossed and a single six-sided die is rolled, describe the sample space.
\(\quad\) b) Find the probability of getting heads and an even number.
\(\quad\) c) Find the probability of getting heads and a number greater than 4
\(\quad\) d) Find the probability of getting tails and an odd number.
3) When rolling a single, six-sided die, find the probability of:
\(\quad\) a) rolling a six
\(\quad\) b) rolling an even number
\(\quad\) c) rolling a number greater than 5
4) When rolling a single, six-sided die, find the probability of:
\(\quad\) a) rolling a two or a three
\(\quad\) b) rolling an odd number
\(\quad\) c) rolling a number divisible by three
5) If a card is drawn randomly from a standard 52 card deck, find the probability of:
\(\quad\) a) drawing a king
\(\quad\) b) drawing a face card
\(\quad\) c) drawing a card that is not a face card
6) If a card is drawn randomly from a standard 52 card deck, find the probability of:
\(\quad\) a) drawing a heart
\(\quad\) b) drawing a heart or a spade
\(\quad\) c) drawing a card that is a heart, a diamond or a spade
7) A ball is drawn randomly from an urn that contains 8 balls - five red, two white and one yellow. Find the probability that:
\(\quad\) a) a red ball is chosen
\(\quad\) b) a yellow ball is not chosen
\(\quad\) c) a green ball is chosen
8 , A ball is drawn randomly from an urn that contains 8 balls - five red, two white and one yellow. Find the probability that:
\(\quad\) a) the ball drawn is neither yellow nor white
\(\quad\) b) the ball drawn is either white, yellow, or red
\(\quad\) c) the ball chosen is not white
9) A drawer contains 18 socks of which 6 are red, 4 are white and 8 are black.
\(\quad\) a) If one sock is drawn at random from the drawer, what is the probability that it is red?
\(\quad\) b) If one red sock is drawn with the first choice, what is the probability that the next sock drawn is also red?
A poker hand is drawn at random from a standard deck of 52 cards.
10) Find the probability of getting five hearts.
11) Find the probability of getting five cards of the same suit.
12) Find the probability of getting five face cards.
13) Find the probability of getting ace, king, queen, jack and ten of the same suit.
14) For problem #14 refer to the sample space in Section 4.1 (or create your own
If a pair of standard six-sided dice are rolled, what is the probability of:
\(\quad\) a) rolling a 7
\(\quad\) b) rolling a 9
\(\quad\) c) rolling doubles (the same number on each die)
\(\quad\) d) not rolling doubles
\(\quad\) e) rolling 9 or higher

SET II
An unbiased coin is tossed 5 times. Find the probability that:
15) the coin lands heads five times
16) the coin lands heads exactly once
17) the coin lands heads at least once
18) the coin lands heads more than once
Two cards are drawn without replacement from a standard deck of 52 cards. Find the probability that:
19) a pair is drawn
20) a pair is not drawn
21) two black cards are drawn
22) two cards of the same suit are drawn
A jar contains three yellow balls and five red balls. If four balls are drawn at random (without replacement), find the probability that:
23) two of the balls are yellow and two are red
24) all of the balls are red
25) exactly three of the balls are red
26) two or three of the balls are yellow
Assume that the probability of a boy being born is the same as the probability of \(a\)
girl being born. Find the probability that a family with three children will have:
27) two boys and one girl
28) at least one girl
29) no boys
30) the two oldest children are girls
31) An exam consists of ten True-or-False questions. If a student guesses at every answer, what is the probability that he or she will answer exactly six questions correctly?
32) A law firm employs 14 lawyers, 8 of whom are partners in the firm. If a group of 3 lawyers is chosen at random to attend a conference, what is the probability that 3 partners will be selected?
33) In a lot of 24 computer components, there are four defective components. If two of the components are chosen at random, what is the probability that:
\(\quad\) a) both of the components are defective?
\(\quad\) b) at least one of the components is defective?
34) A barrel of 60 apples contains 4 rotten apples. If 3 apples are selected at random, what is the probability that 1 or more of the apples is rotten?
35) A shelf at the home improvement store contains 80 light bulbs of which 6 are defective. If a customer chooses 2 light bulbs at random, what is the probability that:
\(\quad\) a) both are defective?
\(\quad\) b) at least one is defective?
36) Some computer components are shipped in boxes of 24. Before they are shipped, the quality control inspector randomly chooses 8 components from each box. If any of the 8 components selected are defective, the box is not shipped. What is the probability that a lot containing exactly 2 defective components would be shipped anyway?
37) A business is preparing to choose 12 people to go on a business trip from a group of 100 employees, 60 of whom are women and 40 of whom are men. Suppose that Ed and Mary are both employees and that the 12 people for the trip will be chosen randomly.
\(\quad\) a) What is the probability that Ed will be chosen?
\(\quad\) b) What is the probability that Ed and Mary will both be chosen?
\(\quad\) c) If an equal number of men and women are to go on the trip, what is the probability that Ed will be chosen?
\(\quad\) d) If an equal number of men and women are to go on the trip, what is the probability that Ed and Mary will both be chosen?
38) A company has has 50 sales representatives on staff. There are three sales calls that must be made on the east side of town and five sales calls on the west side. If the sales reps are chosen at random for these eight sales calls:
\(\quad\) a) What is the probability that an individual sales rep will be chosen for any of the eight sales calls?
\(\quad\) b) What is the probability that two sales reps will be selected to make their sales calls on the same side of town.
39) A student studying for a test knows how to do 12 of the 20 problems from the study guide. If the test contains 10 problems chosen at random from the study guide, what is the probability that at least 8 of the problems on the test are problems the student knows how to do?