Sample Space Statistics Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard
Two dice are rolled. (a) How many outcomes are in the sample space? (b) How many outcomes give a sum of 7? (c) How many outcomes give a sum greater than 10? (d) Which sum is more likely: 7 or greater than 10?

Solution

  1. 1
    Step 1: (a) 6×6=366 \times 6 = 36 outcomes. (b) Sum of 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) = 6 outcomes. (c) Sum > 10 means sum = 11 or 12. Sum=11: (5,6),(6,5) = 2. Sum=12: (6,6) = 1. Total = 3 outcomes.
  2. 2
    Step 2: (d) P(sum=7)=636=16P(\text{sum}=7) = \frac{6}{36} = \frac{1}{6}. P(sum>10)=336=112P(\text{sum}>10) = \frac{3}{36} = \frac{1}{12}. Sum of 7 is twice as likely as sum greater than 10.

Answer

(a) 36 outcomes. (b) 6 give sum of 7. (c) 3 give sum > 10. (d) Sum of 7 is more likely (16\frac{1}{6} vs 112\frac{1}{12}).
Understanding the sample space for two dice is fundamental to probability. The 36 equally likely outcomes allow us to compute exact probabilities by counting favourable outcomes. The sum of 7 has the most combinations of any sum, making it the most likely total.

About Sample Space

The sample space is the complete set of all possible outcomes for a probability experiment, listed without repetition. It forms the foundation for every probability calculation because the probability of any event is a fraction of the sample space.

Learn more about Sample Space →

More Sample Space Examples

Example 1 easy

List the sample space for rolling a standard six-sided die and flipping a coin simultaneously.

Example 2 medium

A restaurant offers 3 starters (soup, salad, bread), 4 mains (chicken, fish, beef, pasta), and 2 des

Example 3 medium

A password consists of one letter (A–E) followed by one digit (1–3). (a) List the entire sample spac

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