What traumatic event does Zitkala-Sa describe at the school?

The forcible cutting of her long hair

Why was the cutting of her hair so humiliating to Zitkala-Sa?

Among her people, shingled hair was worn by cowards and mourners

Define conditional probability.

\(P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}\), provided \(P(B)\neq0\); it is the probability of \(A\) given that \(B\) has occurred.

State the multiplication theorem of probability.

\(P(A\cap B)=P(B)\,P(A\mid B)=P(A)\,P(B\mid A)\).

When are two events independent?

When \(P(A\cap B)=P(A)\,P(B)\); equivalently \(P(A\mid B)=P(A)\).

State Bayes' Theorem.

\(P(E_i\mid A)=\dfrac{P(E_i)P(A\mid E_i)}{\sum_j P(E_j)P(A\mid E_j)}\), for mutually exclusive, exhaustive events \(E_j\).

State the Theorem of Total Probability.

For exhaustive, mutually exclusive events \(E_1,\dots,E_n\), \(P(A)=\sum_i P(E_i)P(A\mid E_i)\).

Define a random variable.

A real-valued function defined on the sample space of a random experiment, assigning a number to each outcome.

Write the mean of a random variable \(X\).

\(E(X)=\sum_i x_i\,P(x_i)\), the expected value of \(X\).

State the binomial probability formula.

\(P(X=r)=\binom{n}{r}p^r(1-p)^{n-r}\), \(r=0,1,\dots,n\), for \(n\) independent trials with success probability \(p\).

Give the mean and variance of a binomial distribution.

Mean \(=np\) and variance \(=np(1-p)\).

What conditions define a binomial experiment?

A fixed number \(n\) of independent trials, each with only two outcomes (success/failure) and a constant success probability \(p\).

Write the formula for the theoretical probability of an event E.

\(P(E) = \dfrac{\text{number of outcomes favourable to } E}{\text{total number of equally likely outcomes}}\).

What is the probability of a sure event and of an impossible event?

\(1\) for a sure (certain) event and \(0\) for an impossible event.

What is the range of the probability of any event?

The probability of any event lies between 0 and 1, that is \(0 \le P(E) \le 1\).

What are complementary events, and how are their probabilities related?

An event \(E\) and "not \(E\)" are complementary; \(P(E) + P(\text{not }E) = 1\), so \(P(\text{not }E) = 1 - P(E)\).

A die is thrown once. What is the probability of getting an even number?

Favourable outcomes are 2, 4, 6 (three of them) out of six, so \(P = \dfrac{3}{6} = \dfrac{1}{2}\).

A card is drawn from a well-shuffled pack of 52. What is the probability that it is a king?

There are 4 kings, so \(P(\text{king}) = \dfrac{4}{52} = \dfrac{1}{13}\).

Two coins are tossed together. What is the probability of getting two heads?

The outcomes are HH, HT, TH, TT; only one is favourable, so \(P(\text{two heads}) = \dfrac{1}{4}\).

If the probability of an event is 0.7, what is the probability of its not happening?

\(P(\text{not }E) = 1 - 0.7 = 0.3\).

Probability

Class 12 Mathematics · Mathematics Part II · Chapter 7

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Summary

This chapter develops probability beyond simple counting. Conditional probability \(P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}\) measures the chance of \(A\) given that \(B\) has occurred. The multiplication theorem follows: \(P(A\cap B)=P(B)\,P(A\mid B)\). Two events are independent when \(P(A\cap B)=P(A)\,P(B)\). The Theorem of Total Probability sums the contributions of mutually exclusive, exhaustive events, and Bayes' Theorem reverses conditioning to find \(P(E_i\mid A)\) from the prior probabilities and likelihoods. A random variable assigns a number to each outcome; its probability distribution leads to the mean (expectation) \(E(X)=\sum x_iP(x_i)\) and variance. The binomial distribution, modelling \(n\) independent Bernoulli trials with success probability \(p\), gives \(P(X=r)=\binom{n}{r}p^r(1-p)^{n-r}\).

Conditional probabilityMultiplication theorem and independenceTotal probability and Bayes' TheoremRandom variables and expectationBinomial distribution

Key terms

Conditional probability

\(P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}\), \(P(B)\neq0\).

Independent events

Events with \(P(A\cap B)=P(A)P(B)\).

Total probability

Summing \(P(E_i)P(A\mid E_i)\) over exhaustive events.

Bayes' Theorem

Reverses conditioning to find \(P(E_i\mid A)\).

Random variable

A function assigning a real number to each outcome.

Binomial distribution

\(P(X=r)=\binom{n}{r}p^r(1-p)^{n-r}\).

Extra questions & answers

Try one from this chapter

Conditional probability \(P(A\mid B)\) equals:

Flashcards & interactive practice