Independent and Dependent Events of Three Coins Tossing
Problem 733
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 they are independent or not.
Add to solve laterSponsored Links
Definition of Independence
Recall that events $E$ and $F$ are said to be independent if
\[P(E \cap F) = P(E) P(F).\] Otherwise, they are dependent.
\[P(E \cap F) = P(E) P(F).\] Otherwise, they are dependent.
Solution.
First of all, we have $P(H_1)= P(H_2)= 1/2$. To calculate the probability $P(E)$, note that we have $E = \{\text{hht}, \text{thh}\}$.
Here $\text{hht}$ means that the first and the second coins land heads and the third lands tails. Similarly for $\text{thh}$.
Here $\text{hht}$ means that the first and the second coins land heads and the third lands tails. Similarly for $\text{thh}$.
Thus,
\[P(E)= \frac{2}{8} = \frac{1}{4}.\]
Now we consider intersections of events.
First, since $H_1 \cap H_2 = \{\text{hhh}, \text{hht}\}$, we see that
\[P(H_1 \cap H_2) = \frac{2}{8} = \frac{1}{4} = \frac{1}{2} \cdot \frac{1}{2} = P(H_1)\cdot P(H_2).\] Therefore, the events $H_1$ and $H_2$ are independent.
First, since $H_1 \cap H_2 = \{\text{hhh}, \text{hht}\}$, we see that
\[P(H_1 \cap H_2) = \frac{2}{8} = \frac{1}{4} = \frac{1}{2} \cdot \frac{1}{2} = P(H_1)\cdot P(H_2).\] Therefore, the events $H_1$ and $H_2$ are independent.
Next, as $H_1 \cap E = \{\text{hht}\}$, we have
\[P(H_1 \cap E) = \frac{1}{8} = \frac{1}{2} \cdot \frac{1}{4} = P(H_1) \cdot P(E).\] Hence, the events $H_1$ and $E$ are independent.
\[P(H_1 \cap E) = \frac{1}{8} = \frac{1}{2} \cdot \frac{1}{4} = P(H_1) \cdot P(E).\] Hence, the events $H_1$ and $E$ are independent.
Finally, since $H_2 \cap E = \{\text{hht}, \text{thh}\}$, we have
\[P(H_2 \cap E) = \frac{2}{8} = \frac{1}{4}.\]
\[P(H_2 \cap E) = \frac{2}{8} = \frac{1}{4}.\]
On the other hand, we have
\[P(H_2) \cdot P(E) = \frac{1}{2} \cdot \frac{1}{4} = \frac{1}{8}.\]
It follows that $P(H_2 \cap E) \neq P(H_2) \cdot P(E)$.
Thus we conclude that the events $H_2$ and $E$ are dependent.
Add to solve later
Sponsored Links
More from my site
If At Least One of Two Coins Lands Heads, What is the Conditional Probability that the First Coin Lands Heads?
Two fair coins are tossed. Given that at least one of them lands heads, what is the conditional probability that the first coin lands heads?
We give two proofs. The first one uses Bays' theorem and the second one simply uses the definition of the conditional […]- 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 […]
- Probabilities of An Infinite Sequence of Die Rolling
Consider an infinite series of events of rolling a fair six-sided die. Assume that each event is independent of each other. For each of the below, determine its probability.
(1) At least one die lands on the face 5 in the first $n$ rolls.
(2) Exactly $k$ dice land on the face 5 […]
- What is the Probability that All Coins Land Heads When Four Coins are Tossed If…?
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 […]
- What is the Probability that Selected Coin was Two-Headed?
There are three coins in a box. The first coin is two-headed. The second one is a fair coin. The third one is a biased coin that comes up heads $75\%$ of the time. When one of the three coins was picked at random from the box and tossed, it landed heads.
What is the probability […]
- 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 […]
- Probability that Alice Wins n Games Before Bob Wins m Games
Alice and Bob play some game against each other. The probability that Alice wins one game is $p$. Assume that each game is independent.
If Alice wins $n$ games before Bob wins $m$ games, then Alice becomes the champion of the game. What is the probability that Alice becomes the […]
- Probability that Alice Tossed a Coin Three Times If Alice and Bob Tossed Totally 7 Times
Alice tossed a fair coin until a head occurred. Then Bob tossed the coin until a head occurred. Suppose that the total number of tosses for Alice and Bob was $7$.
Assuming that each toss is independent of each other, what is the probability that Alice tossed the coin exactly three […]
