Algebra 2 Honors: Topic 12 Probability

12.3A - Counting Problems

Permutation: an ordered arrangement of r objects chosen from n objects

Formula:

  • No repetition: nPr = n!(n−r)!\frac{n!}{\left(n-r\right)!}

  • With repetition: nPr = nr

Ex: How many ways can 7 letters be arranged to form ordered codes of 5 letters?

5 letters are being picked from 7, so r = 5 and n = 7

If the problem says OR, you ADD the probabilities. If it’s AND, MULTIPLY.

12.3B - Permutations

The number of permutations P of n objects taken n at a time, with r objects alike, s of another kind alike, and t of another kind alike is n!r!s!t!\frac{n!}{r!s!t!}

Ex: How many different words can be formed using all letters in MISSISSIPI?

  • There’s 4 S’s, 4 I’s, and 2 P’s, so the equation would be 11!4!⋅4!⋅2!\frac{11!}{4!\cdot4!\cdot2!}

12.3C - Combinations

Combination: a set of n objects in an arrangement without regard to order

Formula: nCr = (nr)=n!(n−r)!⋅r!\left(\frac{n}{r}\right)=\frac{n!}{\left(n-r\right)!\cdot r!}

Ex: From 52 cards, 5 cards are dealt. How many ways can you get 2 aces and 3 kings.

  • You need to find the probability that you’ll get 2 Aces out of the 4 in the deck AND 3 Kings out of the 4 in the deck.

  • So the equation goes (42)⋅(43)=24\left(\frac42_{}\right)\cdot\left(\frac43\right)=24

Choosing r objects from n is the same as choosing the (n-r) objects left behind

12.3D - Mixture of Problems

Formulas/Important notes

Probability of All - Probability of none = probability of at least one

12.3E - Binomial Theorem

Series written through summation notation:

(a+b)n = ∫r=0n ⁣(nr)an−r⋅br\int_{r=0}^{n}\!\left(\frac{n}{r}\right)a^{n-r^{}}\cdot b^{r}

Theorem 15-10: The total number of subsets of a set with n members is 2n

12.1A and 12.1B - Probability Equally Likely Outcomes

Ex: I pick 2 magazines at random. Find the probability they’re both sports magazines with 28 sports, 15 news, and 9 travel magazines.

  • Here, you do (probability of getting what you want of 2 out of 28 magazines)/(the number of ways you can pick 2 out of 52 magazines)

  • So P(sports magazine) = (282)(522)=63221\frac{\left(\frac{28}{2}\right)}{\left(\frac{52}{2}\right)}=\frac{63}{221}

Complement (‘): means NOT

P(D) + P(D’) = 1, and 1 - P(D’) = P(D)

Formulas:

P(A ∪\cup B) (A or B) = P(A) + P(B) - P(A ∩\cap B) (A and B)

REVIEW HOW DO TO VENN DIAGRAMS FOR THESE

12.2A - Conditional Probability

Probability of A Given Probability of B = P(A|B) = P(A∩B)P(B)\frac{P\left(A\cap B\right)}{P\left(B\right)}

  • Here, we’re limiting the sample space to the given

Ex: In a class of seniors, juniors, and sophomores, finding how many are journalism majors GIVEN they’re a junior means we ONLY consider the juniors

12.2B - Independent & Dependent Events

Independent: where two probabilities don’t impact each other

  • Ex: The probability of me going to school doesn’t affect whether my friend goes to school

To find if 2 events are independent, if one of these are true, it’s independent:

  • P(A) = P(A|B)

  • P(B) = P(B|A)

  • P(A ∩\cap B) = P(A) * P(B)

Else, it’s dependent.

12.4A - Binomial Probabilities

Binomial Probabilities: there’s only two outcomes: success or failure

  • it’s a shortcut but only when the probabilities are independent

Formula: P(x) = (nr\frac{n}{r} ) (probability of what you want to happen)The Number of times you want it to happen (n) * (1-probability of what you want to happen)n-r

Ex: On a true/false quiz you guess on 5 questions. What’s the probability you answer 3 questions correctly?

  • The equation is P(c = 3) = (53)(0.5)3(0.5)2=0.3125\left(\frac53\right)\left(0.5\right)^3\left(0.5\right)^2=0.3125

12.5A - Probability Distributions

Probability distributions: rule that assigns a probability to each value of a discrete random value (all probabilities must add up to 1)

  • Look on notes for examples