# What is the Probability that All Coins Land Heads When Four Coins are Tossed If…?

## Problem 730

Four fair coins are tossed.

(1) What is the probability that all coins land heads?

(2) What is the probability that all coins land heads if the first coin is heads?

(3) What is the probability that all coins land heads if at least one coin lands heads?

## Solution.

### Solution (1)

There are $2^4=16$ total possible outcomes of which only one outcome gives rise to all heads.

Thus the probability that all coins land heads is $1/16$.

### Solution (2)

Consider the event that the first coin is heads.

In this case, there are total $2^3=8$ possible outcomes for the rest of coins (2nd, 3rd, and 4th).

Hence, the probability that all coins land heads given that the first coin is heads is $1/8$.

### Solution (3)

Let $H$ be the event that all coins land heads. Let $F$ be the event that at least one coin lands heads. Then the required conditional probability is given by
\begin{align*}
P(H \mid F) &= \frac{P(H \cap F)}{P(F)}.
\end{align*}
The complement $F^c$ of $F$ is the event that all lands tails whose probability $P(F^c)$ is $1/16$ just like part (a). Hence
$P(F) = 1 – P(F^c) = 1 – \frac{1}{16} = \frac{15}{16}.$

It follows that
\begin{align*}
P(H \mid F) &= \frac{P(H \cap F)}{P(F)}\\[6pt] &= \frac{P(H)}{P(F)}\\[6pt] &= \frac{1/16}{15/16}\\[6pt] &= \frac{1}{15}.
\end{align*}

### More from my site

• Independent and Dependent Events of Three Coins Tossing Suppose that three fair coins are tossed. Let $H_1$ be the event that the first coin lands heads and let $H_2$ be the event that the second coin lands heads. Also, let $E$ be the event that exactly two coins lands heads in a row. For each pair of these events, determine whether […]
• Complement of Independent Events are Independent Let $E$ and $F$ be independent events. Let $F^c$ be the complement of $F$. Prove that $E$ and $F^c$ are independent as well. Solution. Note that $E\cap F$ and $E \cap F^c$ are disjoint and $E = (E \cap F) \cup (E \cap F^c)$. It follows that \[P(E) = P(E \cap F) + P(E […]
• Jewelry Company Quality Test Failure Probability A jewelry company requires for its products to pass three tests before they are sold at stores. For gold rings, 90 % passes the first test, 85 % passes the second test, and 80 % passes the third test. If a product fails any test, the product is thrown away and it will not take the […]
• Independent Events of Playing Cards A card is chosen randomly from a deck of the standard 52 playing cards. Let $E$ be the event that the selected card is a king and let $F$ be the event that it is a heart. Prove or disprove that the events $E$ and $F$ are independent. Definition of Independence Events […]
• Pick Two Balls from a Box, What is the Probability Both are Red? There are three blue balls and two red balls in a box. When we randomly pick two balls out of the box without replacement, what is the probability that both of the balls are red? Solution. Let $R_1$ be the event that the first ball is red and $R_2$ be the event that the […]
• Probability Problems about Two Dice Two fair and distinguishable six-sided dice are rolled. (1) What is the probability that the sum of the upturned faces will equal $5$? (2) What is the probability that the outcome of the second die is strictly greater than the first die? Solution. The sample space $S$ is […]
• Conditional Probability Problems about Die Rolling A fair six-sided die is rolled. (1) What is the conditional probability that the die lands on a prime number given the die lands on an odd number? (2) What is the conditional probability that the die lands on 1 given the die lands on a prime number? Solution. Let $E$ […]
• Overall Fraction of Defective Smartphones of Three Factories A certain model of smartphone is manufactured by three factories A, B, and C. Factories A, B, and C produce $60\%$, $25\%$, and $15\%$ of the smartphones, respectively. Suppose that their defective rates are $5\%$, $2\%$, and $7\%$, respectively. Determine the overall fraction of […]

#### You may also like...

This site uses Akismet to reduce spam. Learn how your comment data is processed.

##### Pick Two Balls from a Box, What is the Probability Both are Red?

There are three blue balls and two red balls in a box. When we randomly pick two balls out of...

Close