Probability & Combinatorics Flashcards
There are:
3 green marbles
2 red marbles
1 yellow marble
What is the probability of choosing a green OR yellow marble?
What does OR translate to?
There are 6 total marbles, so denominator is 6
OR translates to +
So, we add: 3/6 chance of green + 1/6 chance of yellow = 4/6 = 2/3
There are:
3 green marbles
2 red marbles
1 yellow marble
If we choose 2 marbles without replacement,
what is the probability of choosing a green AND a red?
What does AND translate to?
AND means we multiply the individual probabilities
There are 2 ways to get a green and a red:
A) 1st marble green, 2nd marble red:
3/6 * 2/5 = 1/2 * 2/5 = 1/5 (once we’ve picked one, there’s only 5 left)
B) 1st marble red, 2nd marble green:
2/6 * 3/5 = 1/3*3/5 = 1/5
Overall, we have (Green AND Red) OR (Red AND Green), so we add:
1/5 + 1/5 = 2/5
There are:
3 green marbles
2 red marbles
1 yellow marble
If we choose 2 marbles without replacement,
what is the probability of choosing 2 greens?
Choosing 2 greens means Green AND Green
AND means we multiply the individual probabilities
3/6 * 2/5 = 1/2 * 2/5 = 1/5
(after we’ve chosen one green, there’s only 2 out of 5 left)
There are:
3 green marbles
2 red marbles
1 yellow marble
If we choose 2 marbles with replacement,
What is the probability of choosing a yellow first, and then a red?
This one is with replacement, so the denominator stays at 6.
There is only 1 option, since it says yellow is first. (we can’t have red, then yellow)
1/6 * 2/6 = 1/18
There are 3 dogs and 2 cats.
If two are chosen at random, what is the probability that they will both be the same type?
Translate: AND is *, OR is +
2 ways: both dogs, OR both cats
(Dog AND Dog) OR (Cat AND Cat)
3/5 * 2/4 + 2/5 * 1/4 = 3/10 + 1/10 = 4/10
If we flip a coin 4 times, what is the probability of it landing on the same side on all 4 flips?
We could have all heads, OR all tails, so we add these 2 probabilities:
A) All heads: 1/2 * 1/2 * 1/2 * 1/2 = 1/16
B) All tails: 1/2 * 1/2 * 1/2 * 1/2 = 1/16
1/16 + 1/16 = 1/8
If there is a 40% chance of rain,
What is the probability that it will NOT rain on the first day, and rain on the second day
Express as a fraction
Probability of NOT A = 1 - Probability of A
Probability of NOT rain = 1 - 40% = 1 - 2/5 = 3/5
AND translates to multiply:
Not Rain * Rain = 3/5 * 2/5 = 6/25
There are 2 sets of cards with numbers:
Set A {1, 2, 4, 6, 9} and Set B {1, 2, 3, 5, 6, 8}
If Joe randomly chooses 1 card from each set,
what is the probability that the numbers multiplied together will be an even number?
There are 3 ways to get an even:
Even * Even
Odd * Even
Even * Odd
There’s only 1 way to get odd: Odd * Odd = Odd
So, it is easier to do 1 - Probability of Odd
Set A: 2/5 chance to get Odd (1 or 9)
Set B: 3/6 = 1/2 chance to get Odd (1, 3, or 5)
Chance of Odd AND Odd = 2/5 * 1/2 = 1/5
1 - 1/5 = 4/5
Two dice are rolled. If the sum of the numbers on the dice was 7, what is the probability that one of the dice showed a number greater than 4?
The possibilities are limited to those that give a sum of 7. (“IF the sum…)
Without this condition, there would be 36 total possibilities (6 for first die * 6 for second die)
List the possibilities:
1-6, 2-5, 3-4, 4-3, 5-2, 6-1
There are 6 total –> that’s the denominator.
4 have a number greater than 4.
4/6 = 2/3
How many different license plates can be made,
using 3 DISTINCT letters,
choosing from the letters A, B, C, D, E, F, G ?
Here, the order matters, so we don’t divide by 3! (3 factorial)
Write out 3 “Slots”, one for each letter.
___ ___ ___
There are 7 possibilites for the first slot, then 6 possibilites for the 2nd slot, then 5 for the 3rd slot (because the letters must be distinct).
We multiply them to find the total number of ways.
_7_ * _6_ *_5_ = 210
How many different teams of 3 can be made from 7 people?
35
When we are picking groups, order does not matter, so we need to divide:
(ABC is the same team as ACB, etc)
How many different ways are there to create a code, using 3 DISTINCT digits, using only integers greater than 1 and less than 8?
Order Matters, so don’t divide:
How many integers are greater than 1 and less than 8?
2, 3, 4, 5, 6, 7 —> 6 digits
Slots method —> 3 slots —> _6_ * _5_ *_4_ = 120
How many different codes can be made from the letters A A A B C?
There are 5 letters, and order matters
If they were all different letters, it would be:
5 * 4 * 3 * 2 = 120
When there are repeats, we divide by the # of repeats, Factorial
So, since there are 3 A’s, it is 120 / 3! = 120 / (3*2) = 20
How many different codes can be made from the letters A A A B B C?
There are 6 letters, and order matters
If they were all different letters, it would be:
6 * 5 * 4 * 3 * 2 = 720
When there are repeats, we divide by the # of repeats, Factorial
So, since there are 3 A’s and 2 B’s we need to divide by 3! and by 2!
720 / ( 3! * 2!) = 720 / (3*2*2) = 60
The diagram shows the various paths along which a mouse can travel from point X, where it is released, to point Y, where it is rewarded with a food pellet. How many different paths from X to Y can the mouse take if it goes directly from X to Y without retracing any point along a path?
Permutations (we multiply the number of options for each slot/choice/fork)
There are 3 forks along the path: 2 choices for the first one, 2 for the second and 3 for the third.
Total # of ways is 2*2*3 = 12
There are 6 women and 3 men. How many different teams of 2 women and 2 men are possible?
Count the women pairs and men pairs separately, then multiply for the total.
Women: 6 * 5 / 2 = 15
Men: 3 * 2 / 2 = 3
15 * 3 = 45
There are 9 candidates for 4 positions on a committee. How many different committees are possible?
126
out of 9, pick 4
4 slots on top, then divide by 4!, because the order doesn’t matter.
