=================================================================================

In probability theory, two events A and B are considered independent if the occurrence (or non-occurrence) of one event does not affect the probability of the other event. 

For instance, for the case of coin tosses, we have: 

1. Events A and B: 

   A: Getting heads on the first toss. 

   B: Getting heads on the second toss. 

2. Independence: 

   Since the coin tosses are independent, then the probability of getting heads on the second toss (B) is not affected by the outcome of the first toss (A). 

3. Conditional Probability (P(B | A)): 

   This represents the probability of event B occurring given that event A has already occurred. In this case, since coin tosses are independent, P(B | A) is simply the probability of getting heads on the second toss, which is 1/2. 

4. Calculation of P(A and B): 

   Using the formula for the probability of the intersection of two independent events: 

            P(A and B) = P(A) × P(B∣A) 

   Since P(A)= 1/2​ (the probability of getting heads on the first toss) and P(B∣A)=  1/2​ (the probability of getting heads on the second toss given that heads occurred on the first toss), then we get,

             P(A and B)= 1/2 × 1/2 = 1/4 

5. Multiplication Rule for Independent Events: 

   This rule states that for independent events, the probability of both events occurring is the product of their individual probabilities: 

            P(A and B) = P(A) × P(B) 

   In this case, since A and B are independent for the case of coin tosses, P(A and B) = 1/2​ × 1/2​ = 1/4. 

Therefore, the calculation P(A and B) =  1/4​ is consistent with the fundamentals of probability for independent events. 

For the case of drawing cards from 52 playing cards, where there are four suits (hearts, diamonds, clubs, spades), and each suit has 13 ranks (numbers 2 through 10, and then jack, queen, king, and ace), assume we have two events (Events A and B): 

A: Drawing a king on the first draw. 

B: Drawing another king without replacing the first card.    

Then, we have, 

1. Dependence:

   These events are dependent because the outcome of event A (drawing a king on the first draw) affects the probability of event B (drawing another king without replacement). 

2. Calculation of P(A and B): 

   To find the probability of both events A and B occurring, we use the multiplication rule for dependent events: 

            P(A and B) = P(A) × P(B∣A) 

   where,

            P(A) is the probability of drawing a king on the first draw, which is 4/52 (since there are 4 kings in a standard deck of 52 cards). 

            P(B∣A) is the conditional probability of drawing another king given that a king was already drawn on the first draw. Since the first king is not replaced, there are now 51 cards left in the deck, and only 3 of them are kings. Therefore, P(B∣A)= 3/51. 

3. Calculation: 

            P(A and B) = (4/52) × (3/51) ​ 

Then, we can simplify this expression to get the probability of drawing a king on both draws without replacement. This calculation takes into account the dependence of events A and B, where the outcome of the first draw affects the probability of the second draw.

============================================