Experimental probability is 3/12 = 1/4 = 0.25, while theoretical probability is 1/6 (about 0.167). They differ due to a small trial sample; with 6000 rolls, experimental probability approaches theoretical probability by the Law of Large Numbers.
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When rolling a 6-sided die, sample space is {1, 2, 3, 4, 5, 6}, giving 6 possible outcomes. The even numbers are 2, 4, and 6, so probability is 3/6 = 1/2 = 0.5.
For 100 tosses yielding 7 bottom landings, 18 top landings, and 75 side landings, the experimental probabilities are P(bottom) = 7/100 = 0.07, P(top) = 18/100 = 0.18, and P(side) = 75/100 = 0.75.
In an illustrative 20-toss trial getting 11 heads and 9 tails, experimental probability of heads is 11 / 20 = 0.55. For the next toss, because a coin has no memory, tails probability remains 1 / 2 = 0.5.
The probability for Arts Club is 11 / 40 = 0.275 (27.5%). The sample proportion for Sports Club is 9 / 40. Across 800 students, the estimated count is (9 / 40) x 800 = 180 students.