Question

A coin is tossed 3 times. Find the probability that all are heads, (a)if it is known that the first is heads, (b) if it is known that the first 2 are heads, (c) if it is known that 2 of them are heads.

Short answer

(a) The probability that all are heads if the first is known to be heads is \(\frac{1}{4}\). (b) The probability that all are heads if the first 2 are known to be heads is \(\frac{1}{2}\). (c) The probability that all are heads if 2 of them are known to be heads is \(\frac{1}{3}\).

Step-by-step solution

Case (a): If it is known that the first is heads

Given the first coin is a head, we are asked to find the probability of all coins being heads. Since the first coin is already a head, we need to find the probability of the second and third coins also being heads. There are three tosses, and each toss has two possible outcomes (heads or tails). Therefore, the total number of outcomes for three tosses is 2^3 = 8. Since the first coin is heads, we now have only 2^2 = 4 possible outcomes for the remaining two tosses. Among these 4 outcomes, there is only one favorable outcome - the second and third coins are also heads (HHH). Hence, the probability that all are heads given the first is a head is \(\frac{1}{4}\).

Case (b): If it is known that the first 2 are heads

Given that the first and second coins are heads, we are asked to find the probability of all coins being heads. Since the first two coins are already heads, we need to find the probability of the third coin being heads. There are two possible outcomes for the third toss - heads or tails. Among these outcomes, there is only one favorable outcome - the third coin is heads (HHH). Hence, the probability that all are heads given the first two are heads is \(\frac{1}{2}\).

Case (c): If it is known that 2 of them are heads

Given that two of the three coins are heads, we need to find the probability of all coins being heads. Let's first find the total outcomes where two of the three coins are heads. There are three such outcomes: HHT, HTH, and THH. Among these outcomes, there is only one favorable outcome - all coins are heads (HHH). Hence, the probability that all are heads given that two of them are heads is \(\frac{1}{3}\).

Key concepts

Conditional Probability

Understanding conditional probability is crucial when trying to navigate problems involving a certain condition or event that has already occurred. It is the probability of an event happening, given that another event has already occurred. In the context of our coin tossing exercise, conditional probability comes into play when we already know the outcome of one or more coin tosses.

For example, case (a) of the exercise considers the probability of getting three heads, given that the first coin toss has already landed on heads. Logically, knowing an outcome has occurred changes the sample space and alters the probability of subsequent events. This consideration is vital in many real-world scenarios, such as predicting weather events, market trends, or even in the strategy of games of chance.

Sample Space

In probability theory, the sample space of an experiment or random trial is the set of all possible outcomes or results of that experiment. When a coin is tossed three times, the sample space is composed of all the different sequences of heads (H) and tails (T) that could result. There are eight possible outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT.

Understanding the sample space is essential because it determines the total number of possible outcomes, which is used to calculate the probability of a particular event. When a condition changes the situation, as with the given cases in our exercise, the sample space is reduced because certain outcomes are guaranteed, thereby increasing the likelihood of the remaining outcomes.

Combinatorics

Combinatorics is a field of mathematics focused on counting, both as an end in itself and as a tool to solve other problems. When we toss a coin three times, combinatorics shows us the different ways in which the event can occur.

In the case of coin tosses, the basic principle of counting is the multiplication rule, used to find the number of possible outcomes. Since we want to know how many different ways the coin can land, and each toss is independent of the others, we multiply the possibilities for each toss together to get the total number of outcomes for the series of tosses: each coin has 2 outcomes, so with 3 coins, we have 2 x 2 x 2 = 8 total outcomes. This calculation is a fundamental combinatorics approach known as the Counting Principle.

Favorable Outcome

A favorable outcome in probability refers to the specific outcome or set of outcomes of an event that we are interested in. To calculate the probability of an event, we divide the number of favorable outcomes by the total number of outcomes (the sample space size).

In case (a) of the exercise, with the first coin guaranteed to be heads, the favorable outcome is that the second and third coins are heads as well, which can only happen in one specific way: HHH. Thus, there is only one favorable outcome against the remaining outcomes of the reduced sample space. Identifying and counting the favorable outcomes correctly is essential to solving any probability problem efficiently, as it directly applies to the calculation of the probability.