Session Conversions

Solution & Explanation

Part 1: Probability of Both Sessions Converting

Given:

• Two user sessions, each with a conversion probability of 0.5.
• Events are independent.

Objective: Calculate the probability that both sessions result in conversions.

Solution:

To find the probability that both sessions convert, we use the principle of independent events:

• If two events, A and B, are independent, the probability of both events occurring is the product of their individual probabilities:

  $P(A \cap B) = P(A) \times P(B)$

• Let $C_1$ be the event that the first session converts, and $C_2$ be the event that the second session converts.

• Given $P(C_1) = 0.5$ and $P(C_2) = 0.5$, the probability that both sessions convert is:

  $P(C_1 \cap C_2) = P(C_1) \times P(C_2) = 0.5 \times 0.5 = 0.25$

Thus, the probability that both sessions convert is 0.25 or 25%.

Part 2: Expected Number of Conversions in N Sessions

Given:

• N user sessions, each with a conversion probability of q.
• Events are independent.

Objective: Determine the expected number of sessions that convert.

Solution:

Each user session can be modeled as a Bernoulli random variable with a success probability $q$. Let $C_i$ be the random variable indicating whether the $i$-th session converts:

• $P(C_i = 1) = q$

• The expected value of $C_i$ is:

  $E[C_i] = 0 \times (1-q) + 1 \times q = q$

• The expected number of converted sessions can be found by summing the expected values of all sessions:

  $E\left[ \sum_{i=1}^{N} C_i \right] = \sum_{i=1}^{N} E[C_i] = \sum_{i=1}^{N} q = Nq$

Thus, the expected number of sessions that convert is $Nq$.

Alternative Perspective:

The sum of N identical Bernoulli random variables with success probability $q$ follows a binomial distribution with parameters $N$ and $q$. The expected value of a binomial distribution is given by:

• $E[B] = Nq$

Hence, the expected number of converted sessions is $Nq$. This aligns with the earlier result, confirming the consistency of our approach.