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In probability, independent events refer to scenarios where the occurrence of one event does not affect the probability of the other event occurring. This can be thought of like flipping a coin or rolling a dice multiple times, where past outcomes do not influence future ones. In the carnival game described, each ball toss is an independent event.

This means that hitting the target or missing it on one throw does not change the likelihood of hitting or missing on the next throw. The probability remains the same for each independent event. They are unaffected by previous outcomes. This concept is crucial because it simplifies the calculation of probabilities over multiple attempts.

When calculating probabilities for sequences of independent events, such as missing the first and hitting the second, it allows us to simply multiply the probabilities of each individual event happening. For example:

• The probability of missing the target on the first and then hitting on the second toss: The necessary operation is multiplying separate probabilities: \[0.8 \times 0.2 = 0.16\]

The term 'probability of success' is often used in probability theory and statistics to denote the likelihood of achieving a predefined successful outcome. In this exercise, hitting the target can be considered a success.

Knowing the probability of success on a single try helps in determining the probabilities for different scenarios across multiple tries. For example, the probability of success for hitting the target on any given attempt is given as 0.2, or 20%. This is crucial because it forms the base for calculating more complex probabilities when multiple attempts are involved, such as finding the probability of hitting the target on at least one of several tries.

When calculating sequential tries, simply use individual success probabilities for each try. This way, the total likelihood of success over many attempts can be determined. Remember, the probability of success and the probability of failure (missing the target), which is calculated as \(1 - \text{probability of success}\), are complementary.

• Probability of a successful hit on the first toss: \(P(\text{Hit on 1st try}) = 0.2\)
• Probability of failing (or missing) on a single try: \(P(\text{Miss}) = 1 - 0.2 = 0.8\)

Cumulative probability is the probability that a random variable takes on a value less than or equal to a specific value. In the context of the carnival game, this concept helps in calculating the probability of winning a prize up to and including the third try.

By calculating cumulative probability, we determine the likelihood of achieving at least one success in a series of attempts, i.e., hitting the target on the first, second, or third try. It aggregates the probabilities of all possible successful outcomes to provide a single probability value.

To find the cumulative probability of hitting the target and winning a prize by the third try, we add up the separate probabilities of all scenarios where you hit the target:

• Probability of hitting on the first attempt: \(0.2\)
• Probability of hitting on the second attempt after missing the first: \(0.16\)
• Probability of hitting on the third attempt after missing the first two: \(0.128\)
• Combined probability of at least one success up to three tries: \(0.2 + 0.16 + 0.128 = 0.488\)

This cumulative probability tells us there is a 48.8% chance of winning a prize if you have up to three tries.