Game plan 3 · Units 3 and 4

Discrete random variables

Bernoulli trials, binomial distributions and general discrete distributions.

Reach for it when

The First Question found statistics, and the question defines success and failure, says “binomial distribution”, or gives more than two outcomes in a data table. A single trial is Bernoulli. Repeated trials are binomial.

You have a Bernoulli trial when success is clearly defined and the probability of each outcome is fixed. If the trial is repeated, you are in the binomial game plan.

Part 1

Bernoulli and binomial

Both have exactly two outcomes on each trial, success and failure. A Bernoulli variable is one trial. A binomial variable counts the successes over $n$ trials.

Game plan 3.1

Bernoulli distributions

  1. 1Identify the probability of success, $p$, and of failure, $1-p$.
  2. 2Apply the mean, variance or standard deviation formula as asked.

Mean, variance and standard deviation

Mean

$$E(X)=\mu=p$$

Variance

$$\operatorname{Var}(X)=p(1-p)$$

Standard deviation

$$\operatorname{SD}(X)=\sqrt{\operatorname{Var}(X)}$$

The Bernoulli distribution

$x$$0$$1$
$P(X=x)$$1-p$$p$
1 − pp01failuresuccess
With $p=0.3$, the bar at $1$ has height $p$ and the bar at $0$ has height $1-p$. The two bars add to $1$.
Game plan 3.2

Binomial distributions

  1. 1Identify that the problem is binomial.
  2. 2Identify the number of trials $n$, the required successes $r$, the probability of success $p$ and of failure $1-p$.
  3. 3For the mean, variance or standard deviation, apply the formulas below.
  4. 4For a probability, first define it carefully from the language cues.
    • more than $r$ → $P(X>r)$
    • at least or no less than $r$ → $P(X\ge r)$
    • less than $r$ → $P(X\lt r)$
    • at most or no more than $r$ → $P(X\le r)$
    • exactly $r$ → $P(X=r)$
  5. 5Technology-free: apply the formula. It is often quicker to find the probability of the event not happening and subtract it from $1$.
  6. 6Technology-active: use BPD for a $P(X=r)$ situation and BCD for anything else.
  7. 7If you are given the probability, work backwards to find the unknown using the $P(X=r)$ formula.

Mean and variance

Mean

$$E(X)=np$$

Variance

$$\operatorname{Var}(X)=np(1-p)$$

Exact probability (Technology-free)

$$P(X=r)=\binom{n}{r}p^{r}(1-p)^{n-r}$$
012345678910mean = np = 3
$X\sim\text{Bin}(10,\ 0.3)$. The bars are tallest around the mean, $np=3$.

Binomial coefficients

Definition

$$\binom{n}{r}=\frac{n!}{r!\,(n-r)!}$$

Useful identities

$$\binom{n}{0}=\binom{n}{n}=1$$$$\binom{n}{1}=\binom{n}{n-1}=n$$$$\binom{n}{2}=\frac{n(n-1)}{2}$$

Quantity words → probability

Phrase in the questionProbability required
more than$P(X>r)$
at least · no less than$P(X\ge r)$
less than · fewer than$P(X\lt r)$
at most · no more than$P(X\le r)$
exactly$P(X=r)$

In the Topic Guides

Part 2

General discrete random variables

If there are more than two outcomes in a table or data display, you have a general discrete random variable.

Game plan 3.3

General discrete random variables

  1. 1If there are more than two outcomes in a table or data display, you have a discrete random variable.
  2. 2To show a table is a valid distribution, check both: the probabilities add to $1$, and each probability is between $0$ and $1$.
  3. 3For an unknown value in a table, use the fact that the probabilities must add to $1$.
  4. 4For the expected value, multiply each value of $X$ by its probability and add.
  5. 5For the variance, apply the formula below. Find $E(X^{2})$ by squaring your values of $X$, multiplying each by its probability and adding.
  6. 6For the standard deviation, take the square root of the variance.

Variance of a discrete random variable

$$\operatorname{Var}(X)=E(X^{2})-\big(E(X)\big)^{2}$$

Example · A valid distribution

$x$$0$$1$$2$$3$
$P(X=x)$$\frac18$$\frac38$$\frac38$$\frac18$

The probabilities add to $1$ and each one lies in $[0,1]$, so this is a valid distribution.

1/803/813/821/83E(X) = 1.5
The distribution is symmetric, so $E(X)=1.5$ sits halfway between the two tallest bars.

In the Topic Guides

Part 3

Common QCAA language: Discrete random variables

The wording tells you which distribution you have and what to find. Look for these phrases.

Bernoulli and binomial cues

Question typeWhat the question says
Identify a Bernoulli trial“Bernoulli experiment / trial / distribution”, a single action with two outcomes (success / failure, hit / miss, four / not a four).Your move One trial with two outcomes, and the word “Bernoulli”. All you need is $p$ and $p(1-p)$.
Justify a binomial model“State the conditions that make this suitable for a binomial random variable”, “which context is (not) suitable for a binomial model”.Your move List all four conditions: a fixed $n$, independent trials, two outcomes, and the same $p$ each time.
Mean, variance and SD“The mean / expected number”, “the variance”, “the standard deviation” of the distribution.Your move For the mean, variance or SD of a binomial, the formulas are $np$ and $np(1-p)$.
Binomial probability“The probability that … [exactly / at least / at most / no more than / fewer than / more than / between] are successful”.Your move Change words like “at least” or “fewer than” into $\ge$ or $\lt$ first. Then use the formula or your calculator.
Find a missing $n$ or $p$A probability is given and “determine $n$” or “determine $p$”.Your move The probability is given but $n$ or $p$ is unknown. Set up the binomial formula and solve for the missing value.

General discrete random variable cues

Question typeWhat the question says
Properties of a distribution“Show that this is a valid probability distribution”, “determine the value of $k$ so the table represents a probability distribution”.Your move Check two things: every probability is between $0$ and $1$, and together they add to $1$.
Construct a distribution“Construct / complete the probability distribution table for $X$”, “use the spinner / dice / histogram to determine the distribution”.Your move Find every probability from the information, then confirm the total is $1$.
Mean, variance and SD“Determine the expected value $E(X)$”, “the variance $\operatorname{Var}(X)$”, “the standard deviation”, applied as “the expected winnings / number / cost”.Your move Use $\sum x\,p(x)$ and $\sum x^{2}p(x)$, then $\operatorname{Var}(X)=E(X^{2})-\big(E(X)\big)^{2}$.
Modelling with DRVs“The expected profit / winnings”, “is the game fair”, “in the long run …”, a decision between options.Your move Calculate the expected value first. The question then wants you to use it in context.

In the Topic Guides

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