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

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• 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 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 […]
• 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 […]
• 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 […]
• 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 […]
• 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 […]
• 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 […]