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The multiplication rule is essential in calculating the probability of two or more independent events occurring simultaneously. When events are independent, the occurrence of one event does not affect the probability of the other. For instance, in our scenario with the four adults, we are determined to find the probability that the first three prefer online news (O) and the fourth prefers a different source (D). We can represent this event as OOOD. To find the probability of these four independent events occurring in sequence, we can use the multiplication rule. For the given problem, the probability of each specific event (O or D) happening in order is multiplied together:

• First adult: P(O) = 0.4
• Second adult: P(O) = 0.4
• Third adult: P(O) = 0.4
• Fourth adult: P(D) = 0.6

Thus, the probability of OOOD is calculated as: \[ P(OOOD) = P(O) \times P(O) \times P(O) \times P(D) = 0.4 \times 0.4 \times 0.4 \times 0.6 = 0.0384 \] Breaking down these events step by step ensures we understand how to apply the multiplication rule for independent occurrences.

Independent events are those where the outcome of one event does not have any impact on the outcome of another. This is a crucial concept in probability, ensuring that our calculations are accurate when determining the likelihood of combined occurrences. In the given problem, the adults' news preferences can be considered independent events. The choice of each adult does not influence the others. Therefore, when calculating the probability of specific sequences such as OOOD, OODO, ODOO, and DOOO, we treat each preference as an isolated event. By independently assigning probabilities: \[ P(O) = 0.4 \], \[ P(D) = 0.6 \] we can reliably multiply these values according to their sequence to obtain the overall probability of each scenario. The principle of independence simplifies the calculation process and allows us to handle complex probability questions systematically.

A probability distribution maps out the likelihood of different outcomes in a random experiment. In this context, it helps us understand all potential arrangements of adults who prefer online news (O) and those who don't (D). Given that we are looking at four adults, the distribution focuses on the permutations of three O's and one D. There are four possibilities in our case: OOOD, OODO, ODOO, and DOOO. Each of these permutations has the same probability due to their independence and equal conditions.For each arrangement, the probability is calculated as follows:

• OOOD: 0.4 \times 0.4 \times 0.4 \times 0.6 = 0.0384
• OODO: 0.4 \times 0.4 \times 0.6 \times 0.4 = 0.0384
• ODOO: 0.4 \times 0.6 \times 0.4 \times 0.4 = 0.0384
• DOOO: 0.6 \times 0.4 \times 0.4 \times 0.4 = 0.0384

To find the total probability of exactly three adults preferring online news and one preferring differently, we sum these probabilities: \[ P(3O, 1D) = 0.0384 + 0.0384 + 0.0384 + 0.0384 = 0.1536 \]

Binomial probability deals with situations where there are exactly two possible outcomes for each trial. In our problem, the binomial setting is evident as each adult either prefers online news (O) or a different source (D). The probability of success (prefer online news) is 0.4, and the probability of failure (prefer a different source) is 0.6. When trying to find the probability of exactly three successes (O) out of four trials (with success O and failure D), we can use the binomial formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \( n \) is the total number of trials (4), \( k \) is the number of successes (3), \( p \) is the probability of success (0.4), and \( 1-p \) is the probability of failure (0.6). Substituting in the values: \[ P(X = 3) = \binom{4}{3} (0.4)^3 (0.6)^{4-3} = 4 \times 0.064 \times 0.6 = 0.1536 \] This result aligns with our earlier calculation using the multiplication rule. Thus, understanding binomial probability provides a systematic approach to solving problems involving a fixed number of trials and binary outcomes.