Probability

To determine the probability that two balls chosen at random from a bag are of the same color, we solve it as follows:Total number of ways to choose 2 balls out of 8: Number of ways to choose 2 green balls out of 5: Number of ways to choose 2 red balls out of 3: Probability that both balls are of the same color: Thus, the probability that two balls chosen at random are of the same color is .

$ = =6+13-3=16$$ =\frac{8}{13}$

Given the probabilities for each coin: Let be the probability that the first coin shows a tail: To find the conditional probability , where is the event "at least two heads show up", we use the formula: Here, is the probability that the first coin is a tail and at least two heads appear in the three tosses. With the first coin being a tail, we focus on the last two coins having at least two heads, which happens only if both are heads: Thus, the probability of is: Plugging this into the formula for conditional probability gives: Therefore, the conditional probability is:

Probability that engine breaks down, Probability that engine breaks down, Probability that engine breaks down, From these, calculate the probabilities that the engines work: The ship cannot complete the voyage if:All three engines fail.Exactly two engines fail.Case 1: All three engines fail Case 2: Exactly two engines failIf and fail, and works: If and fail, and works: If and fail, and works: Total Probability of Failure: Probability that the Ship Can Complete the Voyage:

We toss5 fair coinsindependently.We are asked for the probability thatat least two headsappear.Step 1. Total possible outcomesEach coin has 2 outcomes → total outcomes = .Step 2. Find the complementIt’s often easier to use the complement: “Fewer than 2 heads” means0 headsor1 head.Step 3. Compute and So, Step 4. Compute the desired probability ✅Final Answer: Correct Option:A