Probability

Step 1: Use the addition rule

$ A A \cap B A \cap \bar{B} P(A \cap \bar{B}) $

Answer: 0.25

Step 1: Understanding the problem.
We are tossing a coin three times and need the probability that NO two successive tosses show the same face. The possible sequences of tosses are: - HTH, THT (both have no successive tosses with the same face).

Step 2: Analyzing the total number of possible outcomes.
The total number of possible outcomes for three tosses of a coin is . The possible sequences are: HTH, THT, HHH, TTT, HHT, HTT, THH, TTH.

Step 3: Calculating favorable outcomes.
The favorable outcomes are HTH and THT. Therefore, the number of favorable outcomes is 2.

Step 4: Calculating probability.
The probability is the ratio of favorable outcomes to total outcomes:

Step 5: Conclusion.
The correct answer is (A) 0.25.

To determine the probability of selecting a card with a number less than 9 from the given deck, follow these steps:

1. Identify the total number of cards: The deck contains 10 cards, numbered from 1 to 10.
2. Identify favorable outcomes: Cards with numbers less than 9 are: 1, 2, 3, 4, 5, 6, 7, and 8. This gives us 8 favorable outcomes.
3. Calculate the probability: Probability is the ratio of favorable outcomes to the total outcomes.
   
4. Verify against the range: The computed probability is 0.8

Thus, the probability of selecting a card with a number less than 9 is 0.8 (to one decimal place).

Given:
Random variable and probabilities:

Step 1 — Find K using total probability:

Step 2 — Compute :
Step 3 — Round to one decimal place:

To find the mean of the given data, we'll use the formula for the mean in a frequency distribution:

We perform the following calculations:

• Calculate for each .

• Calculate the total .
• Calculate the total frequency .
• Substitute these sums into the mean formula:
  

Thus, the mean is 7.5.

The probability of getting heads on either toss is:

Using the formula for the probability of either event occurring:

Thus: