Let's go through each part of the question step-by-step.
(a) In what condition [tex]\(P ( A \cap B )= P ( A ) \times P ( B )\)[/tex]?
The condition [tex]\(P ( A \cap B )= P ( A ) \times P ( B )\)[/tex] holds true if and only if events A and B are independent. In other words, the occurrence of event A does not affect the probability of event B occurring, and vice versa.
(b) Tabulate all the possible outcomes.
When a dice and a coin are tossed simultaneously, we need to consider each face of the dice and each side of the coin. The dice has 6 faces (1, 2, 3, 4, 5, 6) and the coin has 2 sides (Head (H) or Tail (T)).
Therefore, the total possible outcomes are:
| Dice | Coin |
|------|------|
| 1 | H |
| 1 | T |
| 2 | H |
| 2 | T |
| 3 | H |
| 3 | T |
| 4 | H |
| 4 | T |
| 5 | H |
| 5 | T |
| 6 | H |
| 6 | T |
These are all the possible outcomes.
(c) Find the probability of getting a prime number on the dice and head on the coin.
Prime numbers between 1 and 6 are: 2, 3, and 5.
The favorable outcomes for getting a prime number on the dice and a head on the coin are:
- (2, H)
- (3, H)
- (5, H)
There are 3 such favorable outcomes out of the 12 total possible outcomes tabulated above.
So, the probability of getting a prime number on the dice and head on the coin [tex]\( P(\text{Prime and Head}) \)[/tex] is given by:
[tex]\[ P(\text{Prime and Head}) = \frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}} = \frac{3}{12} = 0.25 \][/tex]
(d) How much is the probability of getting a composite number on dice and tail on coin is less than 1?
Composite numbers between 1 and 6 are: 4, and 6.
(Note: 1 is neither prime nor composite)
The favorable outcomes for getting a composite number on the dice and a tail on the coin are:
- (4, T)
- (6, T)
There are 2 such favorable outcomes out of the 12 total possible outcomes.
So, the probability of getting a composite number on the dice and tail on the coin [tex]\( P(\text{Composite and Tail}) \)[/tex] is given by:
[tex]\[ P(\text{Composite and Tail}) = \frac{2}{12} = \frac{1}{6} \approx 0.1667 \][/tex]
To find how much less this probability is than 1, we calculate:
[tex]\[ 1 - P(\text{Composite and Tail}) = 1 - 0.1667 = 0.8333 \][/tex]
Therefore, the probability of getting a composite number on the dice and tail on the coin is 0.8333 less than 1.
