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SPI Mathematics · Probability

Experimental Probability and Expected Counts

From a record of trials, probability is how often the event happened over the number of trials; from a probability, the expected number is the probability times the total.

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Why this matters

Five questions in the bank, all EASY, and each is one multiplication or division: a coin tossed 400 times, tickets in a lottery, ice creams or eggs in a lot. The traps are giving the count of the event when the question asks for the rest (good eggs, not bad ones) and dropping a decimal place.

Concept 1 of 2: Experimental probability from a record of trials

When you cannot list equally likely outcomes, you can still count what actually happened. The share of trials in which the event occurred is its experimental probability.

Definition

  • Experimental probability =number of times the event happenednumber of trials= \dfrac{\text{number of times the event happened}}{\text{number of trials}}.
  • It comes from data, so it need not equal the classical value: 225 heads in 400 tosses gives 916\frac{9}{16}, not 12\frac{1}{2}.
  • The two experimental probabilities of an event and its complement still add to 1.

Experimental probability

P(E)=times E happenedtotal trialsP(E) = \frac{\text{times } E \text{ happened}}{\text{total trials}}

Worked example

A die is thrown 300 times and a 6 comes up 45 times. Find the experimental probability of a 6.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Probability · Experimental Probability and Expected Counts

A coin is tossed 400 times and get the following out- comes. Head: 225 times Tail: 175 times when a coin is tossed at random, the probability of getting a head is ____.

Answering 1/2 for a recorded coin

When the question gives the number of heads in a record of tosses, the answer comes from that record. A fair coin's 12\frac{1}{2} is the classical value, not the experimental one.

Concept 2 of 2: Expected number from a probability

A probability is a share. Multiply the share by the total and you get how many to expect; divide a known number by the total and you get the share back.

Definition

  • Expected number == probability ×\times total.
  • Read which count is asked for: if P(bad) is given and the question wants the good ones, subtract from the total, or multiply by 1−P(bad)1 - P(\text{bad}).
  • Working backwards: tickets held == P(win) ×\times tickets sold.

Expected count

expected number=P(E)×total\text{expected number} = P(E) \times \text{total}

Worked example

The probability that a student is absent on a given day is 0.08. In a class of 50, how many students are expected to be absent?
Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Probability · Experimental Probability and Expected Counts

The probability of getting a bad egg in a lot of 400 eggs is 0.045. The number of good eggs in the lot is:

Giving the count of the event, not the one asked

If P(bad egg) is 0.045 in 400 eggs, the bad eggs number 18. A question asking for the good eggs wants 400−18=382400 - 18 = 382, and 18 is printed as a wrong option.

Dropping a decimal place

0.055×600=330.055 \times 600 = 33, not 3.3 or 330. Check the size: 0.055 is about one in twenty, and one twentieth of 600 is 30.
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Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Watch out for (3)

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