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Dice, coin and card probability calculator

Exact probability of dice sums, coin-flip heads and streaks, drawing special cards from a deck and each 5-card poker hand, as a fraction and 1 in N.

更新于 已验证的示例：9

试一试
概率
概率: 0.166667
最大小数位数：6；取最近值，等距时远离零
Exact fraction
1/6
Chance as 1 in N
1 in 6
As a percentage
16.6667%
Favourable outcomes
6
Equally likely outcomes
36

Rolling 2 dice with 6 sides, the sum is exactly 7 in 6 of the 36 equally likely outcomes: a 16.67% chance.

Sum of 2d6, highlighted: sum exactly 7

2: 0.0280.02823: 0.0560.05634: 0.0830.08345: 0.1110.11156: 0.1390.13967: 0.1670.16778: 0.1390.13989: 0.1110.111910: 0.0830.0831011: 0.0560.0561112: 0.0280.02812
计算方法 S

  1. Equally likely outcomes

    SN=62=36S^N = 6^{2} = 36

  2. Count the sums

    #{sum =7}=6\#\{\text{sum } = 7\} = 6

    Counts come from convolving one die's distribution N − 1 times in exact integers (each new die adds a sliding window of S counts).

  3. 概率

    P=16=0.166666666667P = \frac{1}{6} = 0.166666666667

关于Dice, coin and card probability calculator

Dice sums are counted exactly: one die's outcomes are convolved N − 1 times, so all Sᴺ rolls of N dice with S sides are counted rather than simulated. Heads in coin flips follow the binomial distribution, and runs of heads use Feller's recurrence for success runs. Card draws use the hypergeometric distribution, because each card leaves the deck, and poker hands are counted over all 2,598,960 five-card hands.

The default, two six-sided dice summing to 7, is the most likely total: 6 of the 36 rolls, a chance of 1/6. With three dice, a sum of at least 11 comes up exactly half the time, 108 of 216 rolls, and 3 or more heads in a row turn up in 50.8% of 10-flip sequences.

The results assume fair dice, independent flips and a well-shuffled deck; for a biased coin, enter its chance of heads. Answers are exact fractions unless the numbers grow too large.

计算示例

Two dice sum to 7 (defaults)

Game
Dice
Number of dice
2
Sides per die
6
Sum is
Exactly
Target sum
7
概率
0.166667
Exact fraction
1/6
Favourable outcomes
6
Equally likely outcomes
36

核验来源：6 of the 36 ordered rolls give 7 (standard two-dice table)

Three dice sum to at least 11

Game
Dice
Number of dice
3
Sides per die
6
Sum is
At least
Target sum
11
概率
0.5
Exact fraction
1/2
Favourable outcomes
108

核验来源：Python brute force over all 216 rolls of 3d6: 108 have sum ≥ 11 (symmetry about 10.5)

Four dice sum between 10 and 14

Game
Dice
Number of dice
4
Sides per die
6
Sum is
Between
Lowest sum
10
Highest sum
14
Favourable outcomes
595
Exact fraction
595/1296

核验来源：Python brute force over all 1296 rolls of 4d6

Run of 3 heads in 10 fair flips

Game
Coins
Number of flips
10
Chance of heads
0.5
Heads
A run of heads in a row
Run of at least
3 heads
概率
0.507813
Exact fraction
65/128
Favourable outcomes
520

核验来源：Python brute force over all 2¹⁰ sequences: 520 contain HHH

常见问题

What is the most likely sum when rolling two dice?

7, which 6 of the 36 equally likely rolls produce (1+6, 2+5, 3+4 and their reverses), a probability of 1/6 or 16.67%. The extremes 2 and 12 each need one exact roll, 1/36 or 2.78%. With three dice the peak shifts to 10 and 11, each 27 of 216 rolls, or 12.5%.

What are the odds of flipping heads 10 times in a row?

1 in 1,024 with a fair coin, because each flip halves the chance: (1/2)¹⁰ = 1/1,024, about 0.098%. After 9 heads in a row, the 10th flip is still 50% heads, since flips are independent; expecting tails to be "due" is the gambler's fallacy. Any other specific sequence of 10 flips, such as HTHTHTHTHT, is exactly as unlikely.

What are the odds of each poker hand?

Of the 2,598,960 five-card hands, 4 are royal flushes (1 in 649,740), 624 are four of a kind (1 in 4,165) and 3,744 are full houses (about 1 in 694). One pair turns up in 42.26% of deals, and 50.12% of hands hold nothing better than a high card. These are standard counts for the first five cards dealt, with no draws or community cards.

What is the probability of drawing an ace from a deck?

4/52 = 1/13, about 7.69%, for one card from a full 52-card deck. For at least one ace in a 5-card hand, subtract the chance of none: 1 − C(48, 5)/C(52, 5) = 34.12%. Cards drawn without replacement follow the hypergeometric distribution, so each ace drawn lowers the chance that the next card is an ace.

How likely is a streak of heads in a row?

More likely than most people guess. In 10 fair flips there is a 65/128 = 50.78% chance of at least one run of 3 or more heads, because a run can start at any of the 8 positions from flip 1 to flip 8. Long real sequences therefore contain streaks, and a streak alone is weak evidence that a coin is biased.

“Dice, coin and card probability calculator”有多准确？

准确性取决于输入值和方法的假设。十进制运算使用50位有效数字，但估算、数值方法和源数据的精度可能较低；显示时的舍入并不能消除这些限制。 已按独立来源核验的计算示例：9。 例如，“Two dice sum to 7 (defaults)”根据6 of the 36 ordered rolls give 7 (standard two-dice table)进行核验。

这种方法出自哪里？

Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. — §XIII.7 (success runs) and §II.6 (hypergeometric); Wikipedia — Poker probability: frequency of 5-card poker hands; NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.18 Binomial distribution.

关于此计算器

P(sum=t)=#{rolls with sum t}SN,P(k special)=(Kk)(D−Kn−k)(Dn)P(\text{sum} = t) = \frac{\#\{\text{rolls with sum } t\}}{S^N},\qquad P(k \text{ special}) = \frac{\binom{K}{k}\binom{D-K}{n-k}}{\binom{D}{n}}

来源

  1. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. — §XIII.7 (success runs) and §II.6 (hypergeometric)
  2. Wikipedia — Poker probability: frequency of 5-card poker hands
  3. NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.18 Binomial distribution

已对照来源验证

此计算器包含 9 个已解示例，答案来自独立来源。这些示例会在测试套件中运行，你也可以在此运行验证。

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