Probability › Coins

Equal numbers of heads

Solve the first problem 8 problems

1. Arranging identical coins ●○○○○ 10 ptsSix identical coins are arranged in a row. Find the number of arrangements in which four are heads and two are tails.
2. Two heads and two tails ●○○○○ 10 ptsFour fair coins are tossed simultaneously. What is the probability that exactly two coins show heads and two show tails? Give your answer as a decimal.
3. Equal heads from two coins ●●○○○ 20 ptsCoin $A$ is flipped three times and coin $B$ is flipped four times. Both coins are fair. What is the probability that the number of heads obtained from coin $A$ equals the number of heads obtained from coin $B$? Give your answer as a decimal.
4. Same number of heads for two flips ●●○○○ 20 ptsJackie and Phil have two fair coins and a third coin that comes up heads with probability $\frac{4}{7}$. Jackie flips the three coins, and then Phil flips the three coins. Let $\frac{m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
5. Same heads from one and two coins ●●○○○ 20 ptsKeiko tosses one penny and Ephraim tosses two pennies. All pennies are fair. What is the probability that Ephraim gets the same number of heads that Keiko gets? Give your answer as a decimal.
6. Ten numbered coins sum at least 45 ●●●○○ 50 ptsYou have 10 coins. Coin $i$ shows the number $i$ on one side (for $i = 1, \dots, 10$) and $0$ on the other side. You toss all the coins and add up the values that face up. Find the probability that the sum is greater than or equal to $45$. Give your answer as a decimal.
7. Coin flips moving two numbers together ●●●○○ 50 ptsSteph and Jeff each start with the number 4, and Travis is flipping a coin. Every time he flips heads, Steph replaces her number $x$ with $2x-1$, and Jeff replaces his number $y$ with $y+8$. Every time he flips tails, Steph replaces her number $x$ with $\frac{x+1}{2}$, and Jeff replaces his number $y$ with $y-3$. After some (positive) number of coin flips, Steph and Jeff end up with the same number, which is below 2012. How many times was the coin flipped?
8. Matching head counts with unfair coin ●●●○○ 50 ptsJackie and Phil have two fair coins and a third coin that comes up heads with probability $\frac{4}{7}$. Jackie flips the three coins, and then Phil flips the three coins. Let $\frac{m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

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