Predicting Outcomes

Predicting outcomes means using probability to make a reasonable guess about how many times something will happen if a situation is repeated many times. Instead of guessing randomly, we use a ratio built from the possible results of an event to forecast what is likely to happen in the future.

Every prediction starts with knowing the possible outcomes of an event and how likely each one is. If you already understand how probability is written as a fraction, decimal, or percent, this lesson builds directly on that idea — if you need a refresher first, see Understanding probability.

Once we know the probability of an outcome, we can use it to predict how often that outcome should occur over a set number of trials, such as tosses of a coin, spins of a spinner, or draws from a bag.

The starting point for any prediction is the basic probability formula:

\( P(A) = \dfrac{n_f}{n_t} \)

Here \( n_f \) is the number of favorable outcomes (the outcomes we are interested in) and \( n_t \) is the total number of possible outcomes. Once we know \( P(A) \), we can predict how many times outcome \( A \) should occur in \( n \) trials with:

\( E = P(A) \times n \)

\( E \) is the expected, or predicted, number of times the outcome happens. This is the core idea behind every prediction problem: multiply a probability by a number of trials to forecast a result.

Predictions are usually based on theoretical probability, which comes from counting all equally likely outcomes before anything actually happens, such as figuring out the chance of rolling a 3 on a fair die. When we actually carry out trials and record what happens, that is experimental probability. Practicing with real trials, like flipping coins or spinning a spinner many times, is covered in Simple probability experiments.

A good prediction should get closer to the experimental results as the number of trials increases. If the two stay far apart even after many trials, it is often a sign that the outcomes are not actually equally likely.

A bag contains 4 red marbles and 6 blue marbles, for 10 marbles total. One marble is drawn, its color recorded, and then it is put back before the next draw. Predict how many times red will be drawn in 50 trials.

First find the probability of drawing red: \( P(red) = \dfrac{4}{10} = 0.4 \).

Then predict the number of red draws in 50 trials: \( E = 0.4 \times 50 = 20 \).

So we would predict red to be drawn about 20 times out of 50 draws. This does not mean it will be exactly 20 every time, but 20 is the most reasonable prediction based on the probability.

A spinner is divided into four colored regions with these probabilities: red 40 percent, blue 25 percent, green 20 percent, and yellow 15 percent, shown in the bar model below.

If the spinner is spun 200 times, predict how many times it should land on green. Since \( P(green) = 0.20 \), the prediction is \( E = 0.20 \times 200 = 40 \). We would expect green to come up around 40 times out of 200 spins.

Most real prediction problems are described in words rather than given as a ready-made fraction, so the first job is to translate the situation into a probability before applying \( E = P(A) \times n \). For more practice turning a description into a probability and a prediction, see Probability word problems.

Always confirm that outcomes are equally likely before using theoretical probability. Keep the number of favorable outcomes and total outcomes in the same units, and round a predicted number of outcomes to a sensible whole number, since you cannot have a fractional coin flip or marble draw. Finally, remember that a prediction describes what is likely, not what is guaranteed, so real results will usually vary somewhat from the prediction.

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