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Let's go over the questions from the image you uploaded:

### 1. Probability with favorable and possible outcomes (n(E) = 5, n(S) = 10):
The probability of an event \( E \) is calculated as:
\[
P(E) = \frac{n(E)}{n(S)}
\]
Where \( n(E) \) is the number of favorable outcomes and \( n(S) \) is the total number of possible outcomes.

So, for \( n(E) = 5 \) and \( n(S) = 10 \):
\[
P(E) = \frac{5}{10} = 0.5
\]

### 2. Probability with favorable and possible outcomes (n(E) = 12, n(S) = 18):
Using the same formula:
\[
P(E) = \frac{12}{18} = \frac{2}{3} \approx 0.6667
\]

### 3. Probability of rolling a 3 on a die:
For a fair 6-sided die, the probability of rolling a 3 is:
\[
P(3) = \frac{1}{6} \approx 0.1667
\]

### 4. Probability of rolling a 2 on a die:
Similarly, the probability of rolling a 2 is also:
\[
P(2) = \frac{1}{6} \approx 0.1667
\]

### 5. Probability of rolling an even number on a die:
The even numbers on a die are 2, 4, and 6. So, the number of favorable outcomes is 3:
\[
P(\text{even}) = \frac{3}{6} = \frac{1}{2} = 0.5
\]

### 6. Probability of drawing a Queen from a deck of cards:
There are 4 Queens in a standard 52-card deck. So the probability is:
\[
P(\text{Queen}) = \frac{4}{52} = \frac{1}{13} \approx 0.0769
\]

### 7. Probability of drawing a King from a deck of cards:
Similarly, there are 4 Kings in a deck:
\[
P(\text{King}) = \frac{4}{52} = \frac{1}{13} \approx 0.0769
\]

### 8. Probability of drawing a black Jack:
There are 2 black Jacks (one in spades, one in clubs), so:
\[
P(\text{Black Jack}) = \frac{2}{52} = \frac{1}{26} \approx 0.0385
\]

### 9. Probability of drawing a red Ace:
There are 2 red Aces (one in hearts, one in diamonds):
\[
P(\text{Red Ace}) = \frac{2}{52} = \frac{1}{26} \approx 0.0385
\]

### 10. Probability of getting heads when tossing a coin:
For a fair coin, the probability of heads is:
\[
P(\text{Heads}) = \frac{1}{2} = 0.5
\]

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Do you have any questions or would you like further details on any of these calculations?

Here are 5 additional questions you might explore:

1. How would the probability change if the die was biased and some numbers were more likely to appear?
2. What is the probability of not drawing a Queen from a deck of cards?
3. What is the probability of drawing either a King or a Queen from a deck of cards?
4. How does the probability change when rolling two dice and summing the numbers to get an even total?
5. If you rolled a die 3 times, what is the probability of getting exactly one even number?

**Tip:** In probability, complementary events (like getting heads or not getting heads) sum up to 1, which is useful for calculating the probability of events indirectly.

What is the probability of various events based on dice rolls, card draws, and coin tosses?

Explore the basic concepts of probability through examples involving dice rolls, card draws, and coin tosses. Learn key formulas such as P(E) = n(E)/n(S) and apply them to real-world scenarios. Suitable for Grades 6-8.

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