6.1
Estimate the subjective probability of each event and provide an explanation.
a) At least one severe snow storm will occur within the area this winter.
b) All classes will be cancelled next week.
a) Answers will vary; relatively high
b) Answers will vary; relatively low
A messy drawer contains 4 grey socks, 6 white socks, and 8 black socks. What are the odds in favour of randomly drawing a grey sock?
Let event A be drawing a grey sock
p(A)=4/18
=2/9
Probability of drawing a grey sock
P(A')= 1- P(A)
=1 - 2/9
= 7/9
Using definition of odds
odds in favour of A= P(A)/P(A')
= 2/9 ÷ 7/9
=2/7
A fruit basket contains five red apples and three green apples. Without looking you randomly select two apples. What is the probability that you will select two red apples.
Let R= two red apples are selected
n(R)= 5C2 n(S)=8C2
P(R)= n(R)/n(S)
= 10/28 =5/14 =35.7%
The probability of the Leafs winning the Stanley Cup is 3/8, while the probability that the Raptors win the Larry O'Brien trophy is 1/5. What is the probability that both Toronto teams win their trophy?
P(A)= 3/8
P(B)= 1/5
P(A and B)= 3/8 x 1/5
= 3/40
=8%
One card is being chosen from the deck of 52 cards. What is the probability that it is a King or Jack?
Let event A be getting a King
Let event B be getting a Jack
P(A)=4/52 = 1/13 P(B)=4/52 =1/13
P(A or B)= P(A)+P(B)
= 1/13 + 1/13
= 2/13
H<th 1/4 = 25%
T<th
If the chance of a snowstorm in Brampton, Ontario in January is estimated of 0.6, what are the odds against Brampton having a snowstorm in January?
Since P(A)+P(A')=1
odds against A= P(A')/P(A)
=1- P(A)/P(A)
=1-0.6/0.6
=0.4/0.6
=(0.4/0.6)100
=4/6
=2/3
A hockey team has two goalies, six defenders, eight wingers and four centres. If the team randomly selects four centres to attend a charity function, what is the likelihood that they are all wingers?
Let W= they are all wingers
n(W)=8C4 n(S)= 20C4
P(W)=n(W)/n(S)
=70/4845
=14/969 = 1.5%
Terry travels the same route to work every day. He has determined that there is a 0.7 probability that he will have to wait for at least one red light and that there is a 0.4 probability that he will hear his favourite song on his way to work. What is the probability that Terry will hear his favourite song and not have to wait at a red light?
P(A’and B)= P(A’) x P(B)
= (1-0.7)x 0.4
= 0.12
There is a 12% chance that Terry hears his favourite song and doesn’t have to wait for a red light.
Terri attends a fundraiser at which 17 t-shirts are being given away as door prizes. Door prize winners are randomly given a shift from a stock of 3 black shirts, 5 blue shirts and 10 white shirts. Terri really likes the black and blue shirts but is not to keen on the white ones. Assuming that Terri wins the first door prize, what is the probability that she will get a shirt that she likes?
Let event A be that Terri wins a black shirt
Let event B be that Terri wins a blue shirt
P(A)= 3/17 P(B)= 5/17
Terri would be happy if either A or B occurred.
There are 3+5=8 non-white shirts, so
p(A or B)= 8/17
The probability of Terri winning a shirt that she likes is 8/17 or 47%.
Determine the probability of tossing at least one head with three coins.
7/8=88%
Data Management teacher makes an effort to promote good attendance habits, states that the odds of passing her course are 6 to 1 when a student misses fewer than 5 classes. What is the probability that a student with good attendance will pass?
Let event A be that a student with good attendance passes.
since odds in favour of A= P(A)/P(A')
6/1=P(A)/P(A')
=P(A)/1+P(A)
= 6/1+6
=6/7 or 86%
A messy drawer contains three black socks, five blue socks, and eight white socks, none of which are paired up. If the owner grabs two socks without looking, what is the probability that both will be white?
Let A= both socks are white
n(A)= 8C2 n(S)=16C2
P(A)= n(A)/n(S)
=28/120 =23%
A professional hockey team has eight wingers. Three of these wingers are 30 goal scorers. Every fall the team plays an exhibition game against its farm team. In order to make it fairer, the coaches agreed to select two winners at random from the pro club to play for the farm team. What is the probability that the two wingers selected are 30 goal scorers?
P(A)=3/8
P(B|A)= 2/7
p(A and B)= 3/8 x 2/7
= 3/28
There is a 3/28 or 10.7% probability that two professional 30 goal scorers will play for the farm team
A card is randomly selected from a standard deck of cards. What is the probability that either a heart or a face is selected?
Let event A be that a heart is selected
Let even B be that a face card is selected
P(A)= 13/52 P(B)= 12/52
P(A)+P(B)= 13/52+12/52-3/52
=25/52-3/52
=11/26
The probability that either a heart or a face card is selected is 11/26.
When a die is rolled 15 times, 3 came up 7 times. Determine the theoretical probability of rolling a 3 with a die.
1/6
The odds of Ceaser hitting a home run is 4:10. What is the probability of Ceaser hitting a home run?
Let event A be the event that Ceaser hits a home run.
h=4 k=10
P(A)=h/h+k
=4/4+10
=4/14
=4/7
=0.57 = 57%
What is the probability that at least two of a group of five friends will have the same birthday?
Let B=friends have the same birthday
P(B')= 364/365 x 363/365 x 363/365 x 362/365 x 361/365
=0.9675
P(B)=1-P(B')
=0.032
=3.2%
Serena’s computer sometimes crashes while she is trying to use her e-mail program, OutTake. When OutTake “hangs” (stops responding to commands) she is usually able to close OutTake without a system crash. She read that the probability of OutTake hanging in any 15 min period is 2.5%, while the chance of OutTake and the operating system failing together in any 15 min period is 1%. If OutTake is hanging what is the probability that the operating system will crash?
P(A)= 2.5%
P(A and B)=1%
P(B |A) = P(A and B)/P(A)
= 1%/2.5%
= 0.4
There is a 40% chance the operating system will crash when OutTake is hanging
An electronics manufacturer is testing a new product to see whether it required a surge projector. The tests show that a voltage spike has a 0.5% probability of damaging the products power supply, a 0.9% probability of damaging downstream components and a 0.4% probability of damaging both the power supply and other components. Determine the probability that a voltage spike will damage the product.
Let A be damage to the power supply
Let C be the damage to the other component
P(A or C)= P(A)+P(C)-P(AnC)
= 0.5+0.9+0.4
=1%
There is a 1% probability that a voltage spike will damage the product.
A bag of marbles contains five whites, eight blacks and four blues. Determine the probability that a randomly drawn marble is a marble that is not black.
Let A= marble drawn is not black
n(A)= 5+4= 9
n(S)= 5+4+8=17
P(A)= n(A)/ N(S)
= 9/17
=0.529
=52.9% or 53%
The odds of Kristine being on time for work is 7:2. What is the probability that she will make it to work before 8:00 am.
Let A be the event that Kristine arrives on time.
P(A)= 7/7+2
=7/9 = 78%
A hockey team has two goalies, six defenders, eight wingers and four centres. If the team randomly selects four centres to attend a charity function, what is the likelihood that no goalies or centres are selected?
Let G= no goalies or centres are selected.
n(G)=14C4 n(S)20C4
P(G)=n(G)/n(S)
=1001/4845 =20.6%
At the Tire Store, 5 out of every 50 tires is defective. If you purchase 4 tires for your vehicle and they are randomly selected from a set of 50 newly shipped tires, what is the probability that none of the four tires are defective? (Once chosen, the tires are not replaced)
P(tire #1)= 45/50 = 9/10
P(tire #2)= 44/49
P(tire #3)= 43/48
P(tire #4)= 42/47
P(tire #5)= 41/46
9/10 x 44/49 x 43/48 x 42/47 x 41/46
=715176/1105440
= 65%
Mia attends a party, at which they are giving away 25 glow sticks. Winners are randomly given a glow stick from a stock of 7 yellow, 10 blue and 8 green glow sticks. Assuming Mia wins, what is the probability that she will get her favourite colour green and yellow?
Let event A be the event that Mia wins a green glow stick.
Let event B be the event that Mia wins a yellow glow stick.
P(A)= 8/25 P(B)=7/25
There are 8+7=15 non-blue glow sticks, so
P(A or B)=15/25 =3/5
The probability of Mia winning a yellow/green glow stick is 3/5 or 60%.