Probability

An even chance means the probability of an event occurring is equal to the probability of it not occurring, which is 50 percent.

Each coin has 2 possible outcomes (Heads or Tails). For two coins, the total outcomes are 2 * 2 = 4 (HH, HT, TH, TT).

The probability of getting a head on the first toss is 1/2, on the third is (1/2)^3, on the fifth is (1/2)^5, and so on. This is a geometric series with first term a = 1/2 and common ratio r = 1/4. The sum is a/(1-r) = (1/2)/(1 - 1/4) = (1/2)/(3/4) = 2/3.

Probability of first card being a jack = 4/52. Probability of second being a king (given first was jack) = 4/51. Total probability = (4/52) * (4/51) = 16 / 2652 = 4 / 663.

There are 4 queens in a standard deck of 52 cards. The probability of drawing the first queen is 4/52, and without replacement, the probability of drawing a second queen from the remaining cards is 3/51. Multiplying these probabilities gives (4/52) * (3/51) = 1/221.

When 3 coins are tossed, there are 2^3 = 8 possible outcomes. Only one outcome (T, T, T) results in 3 tails. Thus, the probability is 1/8.

When tossing 3 coins, there are 2^3 = 8 possible outcomes. The only outcome with 3 heads is (H, H, H), so the probability is 1/8.

The probability of picking a black ball is 2/10 = 0.2. While 'unlikely' is a subjective description, in the context of probability questions, it is the only qualitative descriptor provided that fits a low probability event.

The total outcomes when rolling two dice are 36. The outcomes where exactly one die shows 5 are (5,1), (5,2), (5,3), (5,4), (5,6), (1,5), (2,5), (3,5), (4,5), (6,5). There are 10 such outcomes, so the probability is 10/36 = 5/18.

Area of rectangle = 50 * 40 = 2000. Area of circle = pi * r^2 = 3.14 * (7.5)^2 = 3.14 * 56.25 = 176.625. Probability of landing inside the circle = 176.625 / 2000 = 0.0883. Probability of landing outside = 1 - 0.0883 = 0.9117, which rounds to 0.91.

Total balls = 12. P(Red) = 6/12 = 1/2. P(Not Red) = 1/2. Drawing 3 balls with replacement, the probability of exactly one red is 3C1 * (1/2)^1 * (1/2)^2 = 3 * 1/2 * 1/4 = 3/8.

A die has 6 faces. The probability of rolling a 4 is 1/6. The probability of rolling a 6 is 1/6. Total probability = 1/6 + 1/6 = 2/6 = 1/3.

Probability of Head = 1/2. Probability of 6 on a die = 1/6. Since these are independent events, the probability of both is (1/2) * (1/6) = 1/12.