Question

Suppose we have ten coins which are such that if the \(i\) th one is flipped then heads will appear with probability \(i / 10, i=1,2, \ldots, 10\). When one of the coins is randomly selected and flipped, it shows heads. What is the conditional probability that it was the fifth coin?

Short answer

The conditional probability that the fifth coin was selected, given that it showed heads, is \(\frac{1}{11}\).

Step-by-step solution

Calculate the probability of Intersection (A ∩ B)#

In our problem, A is the event of selecting the fifth coin and B is the event of getting heads. The intersection of events A and B occurs when the fifth coin is selected and it shows heads. Probability of selecting the fifth coin from the ten coins is 1/10. Probability of getting heads from the fifth coin is 5/10. We multiply these two probabilities to get the probability of intersection. P(A ∩ B) = P(A) * P(B|A) = (1/10) * (5/10) = 1/20

Calculate the probability of getting heads (P(B))#

To calculate the probability of getting heads (P(B)), we should consider each coin's probability of showing heads when randomly chosen. The probability of getting heads from each coin can be represented as: P(H1) = 1/10 * 1/10 P(H2) = 1/10 * 2/10 P(H3) = 1/10 * 3/10 ... P(H10) = 1/10 * 10/10 The total probability of getting heads (P(B)) is the sum of the probabilities of getting heads from each coin: P(B) = P(H1) + P(H2) + P(H3) + ... + P(H10)

Calculate P(B) - the total probability of getting heads#

We now calculate P(B) using the probabilities of flipping heads for each coin: P(B) = (1/10 * 1/10) + (1/10 * 2/10) + (1/10 * 3/10) + ... + (1/10 * 10/10) Since 1/10 is common in all probabilities, we can write: P(B) = 1/10 * (1/10 + 2/10 + 3/10 + ... + 10/10) We observe that the sum is an arithmetic progression with first term 1/10, common difference 1/10, and last term 10/10. The sum of the arithmetic progression can be calculated using the formula: Sum = (n * (a + l)) / 2 where n is the number of terms, a is the first term, and l is the last term. Here, n = 10, a = 1/10, and l = 10/10. Thus, Sum = (10 * (1/10 + 10/10)) / 2 Sum = (10 * 11/10) / 2 = 11/2 So, P(B) = 1/10 * 11/2 = 11/20

Calculate the conditional probability P(A|B)#

Now, we can calculate the conditional probability by plugging the values of P(A ∩ B) and P(B) into the formula: P(A|B) = P(A ∩ B) / P(B) P(A|B) = (1/20) / (11/20) = 1/11 The conditional probability that the fifth coin was selected, given that it showed heads, is 1/11.

Key concepts

Probability of Intersection

Understanding the probability of intersection is key when dealing with combined events in probability theory. It refers to the likelihood of two events happening at the same time. For instance, if we consider two events A and B, the intersection of A and B (denoted as A ∩ B) occurs only when both A and B occur simultaneously. The probability of this intersection, denoted as P(A ∩ B), can often be found by multiplying the probabilities of the individual events assuming they are independent, that is, P(A ∩ B) = P(A) * P(B).

In a practical situation like flipping a coin from a set, if 'A' represents picking a specific coin and 'B' the event of flipping heads, the intersection is the joint occurrence of picking that coin and getting a heads result. So the probability of picking the fifth coin and flipping heads would be calculated by P(A ∩ B) = P(A) * P(B|A), where P(B|A) is the probability of getting heads given that the fifth coin has been chosen.

Total Probability of Getting Heads

The total probability of getting heads, in the context of flipping coins, can be thought of as the overall chance of flipping heads regardless of which specific coin is selected. To determine this, we consider all possible ways heads can appear and sum their probabilities. Essentially, we are applying the law of total probability, which partitions a probability space into a set of distinct scenarios that can result in the event of interest, in this case, flipping heads.

Each coin has its own probability of showing heads, and when these are combined appropriately, we receive the total probability of the event occurring. Suppose each of the ten coins has a different probability of flipping heads, and each coin is equally likely to be chosen. In such a scenario, the total probability of getting heads is the sum of the individual probabilities of getting heads from each of these ten coins.

Arithmetic Progression

An arithmetic progression is a sequence of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the previous term. This simple yet fundamental concept in arithmetic is incredibly useful for analyzing patterns and solving various problems.

In our coin-flipping scenario, the probabilities of obtaining heads form an arithmetic progression. This is because the probability increments in a constant manner from one coin to the next. We have a first term, a common difference, and it's straightforward to calculate the sum of the sequence with the formula involving the number of terms, the first term, and the last term. This sum is crucial in problems like the one we are considering, as it leads us to the total probability across progressively varying scenarios.

Bayes' Theorem

Bayes' theorem is a powerful tool in probability theory and statistics, enabling us to update our beliefs about the probability of an event based on new evidence. This theorem uses the principles of conditional probability to relate P(A|B) and P(B|A), where P(A|B) is the probability of event A given that B has occurred, and P(B|A) is the probability of event B given that A has occurred.

Considering our coin example, if event A is selecting the fifth coin and B is observing heads, Bayes' theorem helps us reverse the conditions. It lets us move from the likelihood of seeing heads if we chose the fifth coin (P(B|A)), to the likelihood of having chosen the fifth coin if we've observed heads (P(A|B)). By calculating the total probability of heads and the probability of the intersection, we can use Bayes' theorem to infer backwards from the outcome to the cause.