Gacha probability — the chance of at least one hit from a rate and a number of draws

On the left are the Rate (%) and the number of Draws. The three cards on the right are the results, and they change with the numbers on the left. At least one hit is the chance of getting the item one or more times. No hit is its opposite, so the two add up to 100%. Expected hits is the average number of hits over many repeats of the same number of draws.

In this example, a 1% rate and 100 draws give 63.40% for at least one hit, 36.60% for no hit and an expected 1 hit.

The chance of no hit gives the chance of at least one hit

The chance of no hit in one draw is 1 − 0.01 = 0.99. If no draw affects another, the chance of no hit in two draws in a row is 0.99 × 0.99, and in 100 draws in a row it is 0.99 multiplied by itself 100 times.

The opposite of "at least one hit" is "no hit", so the chance of at least one hit is 1 minus the chance of no hit.

Chance of at least one hit = 1 − (1 − rate)^draws

The rate is used as a decimal: 1% is 0.01. The cards on Board 1 multiply the result by 100 to show a percentage. At 100 draws the chance of no hit is 36.60% and the chance of at least one hit is 63.40%, which add up to 100%.

With Draws set to 1, at least one hit is 1% and no hit is 99%. With Draws set to 2, no hit is 0.99 × 0.99 = 0.9801, or 98.01%, and at least one hit is 1.99%.

Two draws at 1% do not add up to 2% because a sum would count the case where both draws hit (0.01 × 0.01 = 0.0001, or 0.01%) twice.

If the rate is 1%, are 100 draws almost certain to hit?

With Draws set back to 100, Board 1 returns to 63.40%. The answer is no. After 100 draws the chance of at least one hit is 63.40%, and in the other 36.60% of cases there is no hit at all.

Rate × draws = 1% × 100 = 1 can look like one hit for certain. That 1 is the expected value: the average number of hits over many repeats of 100 draws. Some repeats give none and some give two or more, and the average comes to 1. The share of repeats with none is the 36.60% shown for No hit on Board 1, and Expected hits on Board 1 is 1.

An expected value of 1 does not make the chance of at least one hit 100%. With Rate set to 50 and Draws to 2, the expected number is still 50% × 2 = 1, but the chance of at least one hit is 75% and the chance of no hit is 25%, because two misses in a row have probability 0.5 × 0.5 = 25%.

The next table lists combinations with an expected 1 hit, set on Board 1.

Expected hits is 1 in every row, but the chance of at least one hit is not the same. The smaller the rate and the more draws, the closer it gets to about 63.2% (in mathematics the limit is 1 − 1/e, about 0.632, where e is the constant 2.718…). Even at 0.1% and 1,000 draws it is 63.23%.

When do 50% and 90% come?

How many draws it takes to reach a given chance can be read from how the chance of no hit shrinks. Each draw multiplies it by 0.99, so it falls by 1%. When it has halved, the chance of at least one hit is 50%; when it is down to a tenth, the chance of at least one hit is 90%.

The Time to double card on Board 2 works out, from a change per period and a multiple, how many periods it takes to get there. Here one period is one draw. With −1 (%) as the change and 0.5 as the multiple, it gives the number of draws until the chance of no hit has halved.

All three cards use a change of −1, with multiples of 0.5, 0.1 and 0.01. The chance of no hit reaches 0.5, 0.1 and 0.01 times its starting value after 68.97, 229.11 and 458.21 draws. In terms of the chance of at least one hit, those are the draws at which it reaches 50%, 90% and 99%. The "rule of 72" in brackets on the first card is a separate rough guide that appears only for 2 or 0.5 times, and it is not used here.

Draws are whole numbers, so Board 1 shows where they land. With Rate set to 1 and Draws to 68 the chance is 49.51%, and at 69 it is 50.02%, so 50% is passed at the 69th draw. At 229 it is 89.99% and at 230 it is 90.09%, so 90% is passed at the 230th. At 459 it is 99.01%.

The next table sets the change on all three cards of Board 2 to the same value. A rate of 0.5% is a change of −0.5.

Halving the rate from 1% to 0.5% takes the 50% point from 68.97 to 138.28 draws, about twice as many. At 2% it is 34.31, about half. In every row the draws to 90% are about 3.3 times the draws to 50%; on the 1% row, 229.11 ÷ 68.97 is about 3.3.

How the chance grows as the draws increase

Board 3 follows the chance of at least one hit as the number of draws moves from 0 to 300. On the left are the Rate, the Draws (a slider) and the Target chance (50%). At the top of the middle column is the chance of at least one hit; below it, Gap to target is that chance minus the target. The sweep card on the right draws the gap as a curve while the draws move. Its Move field is set to Draws.

With 100 draws, the chance of at least one hit is 63.40% and the gap to the target is 13.40%. Pressing Draw turns the gap into a curve for 0 to 300 draws. Where the curve crosses 0 (the dotted line) an orange circle appears and the card reads "Reaches 0 at: 68.9671". The gap is 0 when the chance of at least one hit is exactly 50%, at about 69 draws, which matches the 68.97 on Board 2 to two decimal places.

After the curve is drawn, moving the Draws slider moves a black dot along it. At 69 the gap is 0.02% and the dot is just above the 0 line; at 68 the gap is −0.49% and the dot is below it.

With the target changed to 90 and Draw pressed again, the curve is redrawn and reads "Reaches 0 at: 229.1057". At 230 draws the gap is 0.09%, and at 229 it is −0.01%. The 90% mark is passed at the 230th draw, which agrees with the 229.11 on Board 2 and with the results on Board 1.

The curve rises steeply on the left and flattens toward the right. At 100, 200 and 300 draws the chance of at least one hit is 63.40%, 86.60% and 95.10%. The gain per 100 draws gets smaller: about 63.4 points over the first 100 draws, about 23.2 over the next 100 and about 8.5 over the 100 after that.

The rate changes the shape of the curve too. With the target back at 50, the rate set to 2 and Draw pressed again, the card reads "Reaches 0 at: 34.3098". That matches the 34.31 draws to 50% at a 2% rate in the table for Board 2.

What this calculation does not say

The calculation on the boards assumes the following.

• Each draw is unaffected by the others. After 100 draws without a hit, the chance of a hit on the next draw is still the rate.
• The rate is the number entered, and it does not change as the draws add up.
• There is no rule tied to the number of draws, such as "a hit is certain by a given draw".

If an assumption changes, the calculation changes. For example, if a hit is certain by a given draw, the chance of a hit by that draw is 100%. The calculation then has to be split into cases according to the rule, and the formula on this page alone does not give the answer.

Also, the real rate need not be the number entered. The calculation shows the result if the rate entered is correct. The 63.40% means that if 100 draws were repeated many times, about 63% of the repeats would contain at least one hit. It does not predict the result of any single run.

Read next — Compound interest — the future value from an annual rate and a number of years It uses the same Time to double card to find how many periods an amount growing at a steady rate takes to double.

Related — Percentage change — the new value, and the rise needed to get back It covers the result of applying percentage changes one after another, and a curve drawn with the sweep card.

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