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IMAT Mathematics · Formula sheet

Probability and Statistics formulas

9 formulas and 10 common traps for IMAT Mathematics Probability and Statistics, grouped by subtopic.

Full notes with worked examples

Counting: Arrangements and Choices

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The multiplication principle and arrangements (permutations)

Arrangements

nPr=n!(n−r)!with repeats: n!p! q!⋯{}^nP_r = \frac{n!}{(n - r)!} \qquad \text{with repeats: } \frac{n!}{p!\,q!\cdots}
  • nnnumber of objects available
  • rrnumber placed in order
  • p,qp, qsizes of groups of identical objects

Combinations: choosing a group when order does not matter

Combinations

(nr)=n!r! (n−r)!\binom{n}{r} = \frac{n!}{r!\,(n - r)!}
  • nnnumber of objects to choose from
  • rrnumber chosen

Common traps

Identical letters do not make new arrangements

Swapping two identical letters gives the same word, so n!n! overcounts. Divide by the factorial of the size of each group of repeats; forgetting this gives a much larger number that IMAT puts among the options.

Ask whether order matters before you count

Named posts, rankings, codes and words are ordered: use permutations. Teams, hands of cards and sets of toppings are not: use combinations. The two answers differ by a factor of r!r!, and both appear among the options.

Probability: Outcomes, Rules and Trees

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Probability from equally likely outcomes, and the complement

Equally likely outcomes

P(A)=n(A)n(S)P(not A)=1−P(A)P(A) = \frac{n(A)}{n(S)} \qquad P(\text{not } A) = 1 - P(A)
  • n(A)n(A)number of outcomes in the event
  • n(S)n(S)number of possible outcomes

The addition rule for 'A or B'

Addition rule

P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)
  • A∪BA \cup BA or B (or both)
  • A∩BA \cap BA and B together

Independent events and tree diagrams (with replacement)

Independent events

P(A∩B)=P(A) P(B)P(A \cap B) = P(A)\,P(B)

Conditional probability and drawing without replacement

Conditional probability

P(B∣A)=P(A∩B)P(A)P(B \mid A) = \frac{P(A \cap B)}{P(A)}

Common traps

With two dice, (2, 5) and (5, 2) are different outcomes

The 36 ordered pairs are equally likely; the 21 unordered pairs are not (a double can happen one way, a mixed pair two ways). Count ordered pairs, so a mixed pair counts twice and a double once.

Subtract the overlap when events can happen together

P(A)+P(B)P(A) + P(B) alone is right only for mutually exclusive events. If the sum comes out above 1, or the events clearly overlap (a heart and a king), subtract P(A and B)P(A \text{ and } B).

Exactly one needs every order

"Exactly one red in three spins" is three different paths on the tree. Multiplying the probabilities along a single path gives only one of them; multiply by the number of orders.

Without replacement, the second fraction changes

After one draw the bag has one item fewer, so the second denominator drops by one. Keeping the same fraction for both draws answers the with-replacement question instead, and that answer is always among the options.

P(B given A) is not P(A given B)

The probability that a biology student studies chemistry is not the probability that a chemistry student studies biology: the two divide the same overlap by different totals.

Mean, Median, Mode and Spread

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Mean, median, mode and range

Mean

xˉ=∑xn∑x=n xˉ\bar{x} = \frac{\sum x}{n} \qquad \sum x = n\,\bar{x}
  • ∑x\sum xsum of all the values
  • nnnumber of values

Combined and weighted means, and how averages change when data change

Combined mean

xˉ=n1xˉ1+n2xˉ2n1+n2\bar{x} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}
  • n1,n2n_1, n_2sizes of the two groups
  • xˉ1,xˉ2\bar{x}_1, \bar{x}_2their means

Frequency tables and standard deviation (in outline)

Mean of a frequency table and standard deviation

xˉ=∑fx∑fσ=∑(x−xˉ)2n\bar{x} = \frac{\sum f x}{\sum f} \qquad \sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{n}}
  • fffrequency of the value xx
  • σ\sigmastandard deviation

Common traps

Order the data before finding the median

The median is the middle of the ordered list, not the middle of the list as written. With an even number of values it is the mean of the two middle values, which need not be one of the data.

A combined mean is not the average of the two means

Pooling a group of 20 and a group of 30 weights the second more. Work through the totals; the simple average of the two means is right only when the groups are equal in size.

Shifting every value does not change the spread

Adding the same number to every value moves the mean but leaves every distance from the mean the same, so the standard deviation and the range do not change. Only scaling changes the spread.

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