Dependent or Independent
A) Are the following events independent or dependent?
You pick a card out from a standard deck and then flip a coin.
B) What is the probability of picking a red card and getting a heads?
A) Independent
B) 1/4
P(AandB)= P(A) * P(B)= 26/52 * 1/2=26/104=1/4
A card is chosen at random from a pack of 52 playing cards. What is the probability of a King or a Queen?
2/13
A bowl contains four apples, three bananas, three oranges, and two pears. If one piece of fruit is selected at random, what is the probability of selecting an orange?
3/12=1/4
For a group of five people, what are the chances that any of them celebrate their birthday on the same day of the week?
The "no match" case for:
• 2 people is 6/7
• 3 people is (6/7) × (5/7)
• 4 people is (6/7) × (5/7) × (4/7)
• 5 people is (6/7) × (5/7) × (4/7) × (3/7)
So the chance of not matching is:
(6/7) × (5/7) × (4/7) × (3/7) = 0.1499...
Flip that around and we get the chance of matching:
1 − 0.1499... = 0.8500...
So, there is a 85% chance of any of them celebrating their birthday on the same day of the week.
How many permutations of 3 different digits are there, chosen from the ten digits 0 to 9 inclusive?
P(10,3) = 10 x 9 x 8 = 720
The arrangement of objects in a certain order is called a:
a. Permutation b. Combination
a. Permutation
The probability of selecting a red marble, not replacing it, then selecting a green marble from a box of 6 red and 2 green marbles
A) Are these events dependent or independent?
B) Determine the probability
A) Dependent
B) 6/8 * 2/7 = 12/56=3/14
At a local high school, 34% of the students take a bus to school and 56% of the students walk to school.
a) Find the probability of randomly selecting a student that takes a bus or walks to school.
b) Determine if these events are mutually exclusive or overlapping event?
a) 34/100 + 56/100 = 90/100 or 90%
b) mutually exclusive
What is the probability of selecting a yellow or blue marble from a box of 5 green, 3 yellow, and 2 blue marbles?
P(yellow or blue)= P(yellow) + P(blue) = 5/10=1/2
What are the chances that with 7 people any of them celebrate their Birthday in the same month? (Assume equal months)
The "no match" case for:
• 2 people is 11/12
• 3 people is (11/12) × (10/12)
• 4 people is (11/12) × (10/12) × (9/12)
• 5 people is (11/12) × (10/12) × (9/12) × (8/12)
• 6 people is (11/12) × (10/12) × (9/12) × (8/12) × (7/12)
• 7 people is (11/12) × (10/12) × (9/12) × (8/12) × (7/12) × (6/12)
So the chance of not matching is:
(11/12) × (10/12) × (9/12) × (8/12) × (7/12) × (6/12) = 0.11...
Flip that around and we get the chance of matching:
1 − 0.11... = 0.89...
So, there is an 89% chance of any of them celebrating their Birthday in the same month.
How many permutations of 4 different letters are there, chosen from twenty-six letters of the alphabet?
P(26,4) = 26 x 25 x 24 x 23 = 358,800
the set of all possible outcomes of an event is called
sample space or tree diagram
You pick a card out of a deck, put it back, then pick another card
Are these events independent or dependent?
Independent
A card is drawn from a standard deck of cards. What is the probability of drawing a Jack or a Heart?
A) Are these events mutually exclusive or overlapping?
B) Determine the probability of the event
A) Overlapping event
B) 4/52 + 13/52 - 1/52 =16/52= 4/13
What is the probability of rolling two dice and getting a sum greater than 10
(5,6), (6,5), or (6,6)
3/36= 1/12
What are the chances that with 6 people any of them celebrate their Birthday in the same month? (Assume equal months). Give your answer to the nearest 0.1%
The "no match" case for:
• 2 people is 11/12
• 3 people is (11/12) × (10/12)
• 4 people is (11/12) × (10/12) × (9/12)
• 5 people is (11/12) × (10/12) × (9/12) × (8/12)
• 6 people is (11/12) × (10/12) × (9/12) × (8/12) × (7/12)
So the chance of not matching is:
(11/12) × (10/12) × (9/12) × (8/12) × (7/12) =0.222801
Flip that around and we get the chance of matching:
1 − 0.222801 =0.777199 = 77.7%
So, there is a 77.7% chance of any of them celebrating their Birthday in the same month.
How many different committees of 5 people can be chosen from 10 people?
C(10,5) = 10!/(5!)(5!) = 252
Events that can not happen at the same time are
mutually exclusive
You flip a coin and then flip the same coin again.
A) Are these events independent or dependent?
B) Find the probability of getting two heads
A) Independent
B) 1/2 * 1/2= 1/4
On a school board, 2 of the 4 female members are over 40 years of age, and 5 of the 6 male members are over 40. If one person did not attend the meeting, what is the probability that the person was a male or a member over 40?
4/5
P(male or over 40)= P(male) + P(over 40) - P(male and over 40) = 6/10+ 7/10 - 5/10= 8/10= 4/5
A bag contains 3 black, 5 green, and 4 yellow marbles.
What is the probability that 2 marbles selected a random will both be black?
P(2 black marbles)= 3/12*2/11=6/132= 1/22
There are 12 people in a room ... what is the chance that any two of them celebrate their birthday on the same day? Assume 365 days in a year.
The chance of not matching = 364/365 × 363/365 × 362/365 × 361/365 × 360/365 × 359/365 × 358/365 × 357/365 × 356/365 × 355/365 × 354/365 = 0.8329...
And the probability of matching is 1 − 0.8329...
The probability of sharing a birthday = 1 − 0.8329... = 0.1670...Or a 16.7% chance.
Jones is the Chairman of a committee. In how many ways can a committee of 5 be chosen from 10 people given that Jones must be one of them?
C(9,4) = 9!/(4!)(6!)=126
What is an overlapping event?
it is where an event has something in common that you need to subtract.
A box contains 5 green pencils and 7 yellow pencils. Two pencils are chosen at random from the box without replacement. What is the probability they are different colors?
a) P(Yellow Followed by Green) = 7/12 x 5/11 = 35/132
b) P(Green followed by Yellow) = 5/12 x 7/11 = 35/132
c) P(Different Colors) = 35/132 + 35/132 = 70/132 =35/66
In a class of 29 children, 15 like history and 21 like math. They all like at least one of the two subjects.
What is the probability that a child chosen at random from the class likes math but not history?
14/29
Find the probability of randomly selecting 3 red pencils from a box containing 5 red, 3 blue, and 4 green pencils
5/12*4/11*3/10=60/1320=1/22
There are 4 teenagers in a room. Their ages are in the range {13, 14, 15, 16, 17, 18, 19}
What is the chance that any two of them are the same number of years old?
There are seven teenage years (13, 14, 15, 16, 17, 18 and 19).
So,
the chance of not matching = (6/7) × (5/7) × (4/7) = 0.34985...
And the probability of matching is 1 − 0.34985...
The probability of two being the same number of years old = 1 − 0.34985... = 0.65014...Or a 65.0% chance.
An encyclopedia has 10 volumes. In how many ways can the 10 volumes be arranged on the shelf?
10! = 3,628,800
It is a subset of the sample space that describes an occurrence.
Event
There are two traffic lights along the route that Ms. Bella drives from home to work. One traffic light is red 50% of the time. The next traffic light is red 60% of the time. The lights operate on separate timers. Find the probability that these lights will both be red on Ms. Bella's way home from work.
A) Are these events independent or dependent?
B) Determine the probability that both lights will be red
A) Independent
B) 50/100 * 60/100 = 3/10 or 30%
Two fair dice are thrown. What is the probability that the score on the first die is 6 or the score on the second die is 5?
Are these events mutually exclusive or overlapping?
A) 11/36
B) Overlapping events
A survey of the junior class at SPF high school shows that 2/5 of the students who have home computers use them for word processing, 1/3 use them for playing games, and 1/4 use them for both word processing and playing games. What is the probability that a student with a home computer uses it for word processing or playing games?
29/60
P(AorB)= P(A) + P(B) - P(AandB)
P(AorB)= 2/5 + 1/3 - 1/4
= 24/60 + 20/60 - 15/60=29/60
There are 5 people in a room ... what is the chance that any two of them celebrate their birthday on the same day? Assume 365 days in a year.
The chance of not matching = 364/365 × 363/365 × 362/365 × 361/365 = 0.9728...
And the probability of matching is 1 − 0.9728...
The probability of sharing a birthday = 1 − 0.9728... = 0.0271...Or a 2.71% chance.
Assuming that any arrangement of letters forms a 'word', how many 'words' of any length can be formed from the letters of the word FUN?
(No repeating of letters)
The number of one letter word: P(3,1) = 3
The number of two letter words: P(3,2) = 6
The number of three letter words: P(3,3)=6
This is where order is not important
a. Combination b. Permutation
a. Combination
In Exton School, 40% of the girls like music, and 12% of the girls like music and dance. What percent of those that like music also like dance?
P(Dance|Music) = P(Dance and Music)/P(Music) = 0.12/0.4=30%
There are 30 children in a class and they all have at least one cat or dog. 14 children have a cat, 19 children have a dog. What is the probability that a child chosen at random from the class has both a cat and a dog?
1/10
There are 6 people in a room and it's a leap year.
What is the chance that any two of them celebrate their birthday on the same day? (There are 366 days in a leap year.)
The chance of not matching
= 365/366 × 364/366 × 363/366 × 362/366 × 361/366 = 0.95964...
And the probability of matching is 1 − 0.95964...
The probability of sharing a birthday = 1 − 0.9596... = 0.04035...Or a 4.04% chance.
There are 6 people in a room and it's a leap year.
What is the chance that any two of them celebrate their birthday on the same day? (There are 366 days in a leap year.)
The chance of not matching
= 365/366 × 364/366 × 363/366 × 362/366 × 361/366 = 0.95964...
And the probability of matching is 1 − 0.95964...
The probability of sharing a birthday = 1 − 0.9596... = 0.04035... Or a 4.04% chance.
A password consists of three letters of the alphabet followed by three digits chosen from 0 to 9. Repeats are allowed. How many different possible passwords are there?
26 x 26 x 26 x 10 x 10 x 10 =17,576,000
What is the formula for conditional probability:
P(B|A)?
P(B|A)=P(A and B)/P(A)