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Probability with a Single Die

Probability · Class IX

Pick a target value k; see P(roll = k), P(roll <= k), P(roll > k) on a fair six-sided die.

4
type a value
16
Live values
  • P(roll = k)0.1667
  • P(roll <= k)0.6667
  • P(roll > k)0.3333
  • Odds against rolling k5
Live readout
P(roll = k)0.1667
P(roll <= k)0.6667
P(roll > k)0.3333
Odds against rolling k5
Target k4

Formulas in this lab

  • P(roll = k)
    P(X=k)=1/6P(X = k) = 1/6
  • P(roll <= k)
    P(X≤k)=k/6P(X \le k) = k/6
  • P(roll > k)
    P(X>k)=(6−k)/6P(X > k) = (6-k)/6
  • Odds against rolling k
    5:15 : 1
Tip: Every face has probability 1/6 , fair dice have no memory and no bias.

Frequently asked questions

▶How is probability defined?

For an event E, $P(E) = \frac{\text{favourable outcomes}}{\text{total outcomes}}$. For a fair six-sided die, $P(\text{roll} = 4) = \frac{1}{6}$. Every probability lies between 0 (impossible) and 1 (certain).

▶How do I use this lab?

Pick a target k from 1 to 6. The lab shows $P(\text{roll} = k)$, $P(\text{roll} \le k)$, and $P(\text{roll} > k)$ as fractions and decimals. Try k = 4 to see 1/6, 4/6, and 2/6.

▶Common mistake on this topic

Students write probabilities greater than 1 or as percentages without the percent sign. Probabilities are always in $[0, 1]$ as fractions or decimals. Also, $P(\text{roll} \le k) + P(\text{roll} > k) = 1$, not 6/6 plus something extra.

▶Where do we use probability daily?

Weather forecasts ('70 percent chance of rain'), cricket-toss decisions, and Ludo dice rolls all use probability. Card games on Diwali nights are richer when you know which combinations are likely versus rare.