Question

1. given the following table, find:

a. ( p(x < 2) )
b. ( p(x leq 2) )
c. ( p(x geq 1) )
d. ( p(x
eq 4) )
e. ( e(x) )

1. a golfer has a 40% chance of making an 8 foot putt. ( x=# ) of putt attempts before the golfer makes the putt.

a. what is the expected value for ( x ) in this problem?
b. ( p(x < 3) )
c. ( p(x > 5) )

Explanation:

Step1: Calculate \(P(X < 2)\)

For \(P(X < 2)\), we sum the probabilities for \(X = 0\) and \(X = 1\).
\(P(X<2)=P(X = 0)+P(X = 1)\)
\(P(X<2)=0.05 + 0.10\)
\(P(X<2)=0.15\)

Step2: Calculate \(P(X\leq2)\)

For \(P(X\leq2)\), we sum the probabilities for \(X = 0\), \(X = 1\), and \(X = 2\).
\(P(X\leq2)=P(X = 0)+P(X = 1)+P(X = 2)\)
\(P(X\leq2)=0.05+0.10 + 0.25\)
\(P(X\leq2)=0.40\)

Step3: Calculate \(P(X\geq1)\)

We use the formula \(P(X\geq1)=1 - P(X = 0)\)
\(P(X\geq1)=1-0.05\)
\(P(X\geq1)=0.95\)

Step4: Calculate \(P(X

eq4)\)
We use the formula \(P(X
eq4)=1 - P(X = 4)\)
\(P(X
eq4)=1 - 0.20\)
\(P(X
eq4)=0.80\)

Step5: Calculate \(E(X)\)

The formula for the expected value \(E(X)=\sum_{i}x_{i}P(x_{i})\)
\(E(X)=(0\times0.05)+(1\times0.10)+(2\times0.25)+(3\times0.40)+(4\times0.20)\)
\(E(X)=0 + 0.10+0.50 + 1.20+0.80\)
\(E(X)=2.6\)

Answer:

A. \(0.15\)
B. \(0.40\)
C. \(0.95\)
D. \(0.80\)
E. \(2.6\)