Estimate a probability from experimental results, and use a probability to predict how many times an outcome should happen in a set number of trials.
The July fair is going up on the field. Dee is testing the hoop-la stall before it opens: throw a wooden hoop, and if it drops cleanly over a peg you win. No model here says how likely a clean drop is, and a hoop has no matching sides to argue from the way a coin or a die does. So she throws. After 120 practice throws, 18 hoops landed on a peg. The estimate is successes divided by trials, 18 / 120 = 0.15, and it is called the relative frequency.
Keep the trials independent, with the same chance each time. Independent means one result does not change another trial's chance. In a run of 20, each result is one twentieth of the data; in 120, it is one hundred and twentieth. Larger samples usually give steadier estimates, but a particular larger sample can be farther away. No successes in twenty tries does not prove an event impossible. To pool runs under the same conditions, add their successes and trial counts, then divide once.
Once you have a probability you can run it forwards. If a throw lands on a peg with probability 0.15, then in a long afternoon of 400 throws you would expect 0.15 × 400 = 60 winners. That is the expected number: multiply the probability by the number of trials. Call the probability p and the rule is expected number = p × trials. It runs backwards too: divide a known expected count by the trials and p comes back.
Dee orders 60 prizes and Mia asks a good question: what if 67 people win? The 60 is a centre, not a promise. An afternoon of 400 throws could easily hand out 52 prizes or 69, in the same way a fair coin tossed 400 times rarely lands on exactly 200 heads. So Dee orders 60 and a few spares. Each throw is its own event, so a run of losses never makes the next hoop more likely to land.
Expect 60, prepare for anything near 60.
A relative frequency is a probability estimate from observed trials. A different run can give a different estimate; an exact match to the true probability can also occur without being known. A fair-die model gives each face probability one sixth. Without a suitable model here, Dee uses trials. The estimate describes the tested conditions. If she moves the pegs, she cannot assume the old estimate still applies.
Relative frequency is successes divided by trials. With independent trials and a fixed chance, larger samples usually give steadier estimates, but one larger sample can be farther away. Multiply a probability by the trial count to find the expected number of successes. The actual count can differ. After conditions change, the old estimate cannot be assumed to apply.
Flip a fair coin ten times and you might get seven heads, but over a thousand flips the share usually sits nearer a half. This page runs coin and dice experiments to show the gap between short run wobbles and long run behaviour. Try the flips yourself and compare your fractions with a friend.
The habit to build is always dividing outcomes by trials. That one fraction turns raw tallies into probabilities you can actually use for predictions.
The expected number of successes is probability times trials: the long run average you would predict. This page works through spinners and board games to show how one multiplication forecasts whole seasons of results. Run the examples with your own numbers to feel the scaling.
Note the honesty built into the idea. Expecting 60 reds in 200 spins does not promise exactly 60, it names the centre the results cluster around.
This section builds probability models from scratch: equally likely outcomes, complements, and the rules for combining events. It supplies the formulas behind every estimate you will make.
Focus on the equally likely outcome examples. Most GCSE probability questions reduce to counting wanted outcomes over total outcomes, exactly as shown here.
The July fair is going up on the field. Dee is testing the hoop-la stall: throw a wooden hoop, and if it drops cleanly over a peg you win. She wants the chance of winning before the gates open, and no model here says how likely a clean drop is, because a hoop is not a coin.
Dee throws 120 practice hoops and 18 land on a peg. What is the estimated probability of a win?
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