Question

likelihood of each value. complete parts a through c below
p(27) = 0.35 (type an integer or a decimal. do not round
b. what is the probability that x equals 24 or 26?
p(24 or 26) = 0.3 (type an integer or a decimal. do not round.)
c. what is p(x ≤ 25)?
p(x ≤ 25) = \square (type an integer or a decimal. do not round
clear all

Explanation:

Step1: Identify known probabilities

From the problem, we know \( p(23) = 0.15 \) (from the bar graph's height for \( x = 23 \)), \( p(24) \) can be found as the remaining part? Wait, no, let's recall that the sum of all probabilities should be 1? Wait, no, first, let's list the given probabilities: \( p(27)=0.35 \), \( P(24 \text{ or } 26)=0.3 \). Wait, actually, from the bar graph, let's assume the heights: for \( x = 23 \), height is 0.15; \( x = 24 \), let's say the height is \( a \); \( x = 25 \), height \( b \); \( x = 26 \), height \( c \); \( x = 27 \), height 0.35. We know that \( P(24 \text{ or } 26)=p(24)+p(26)=0.3 \). Also, we can find \( p(23)=0.15 \), \( p(27)=0.35 \). Let's first find \( p(24) \) and \( p(25) \)? Wait, no, for part c, \( P(x \leq 25)=p(23)+p(24)+p(25) \). Wait, let's check the total probability. Wait, maybe the bar graph: let's see, the y-axis is probability. Let's assume:

From the graph, \( p(23) = 0.15 \), \( p(24) \): let's see, the bar for 24 is shorter, maybe 0.05? Wait, the y-axis has 0.05, 0.10, 0.15, 0.20. Wait, the bar for 24 is at 0.05? Wait, the user's image: "0.05" is a mark, and the bar for 24 is at 0.05? Wait, maybe:

Wait, let's re-express:

Given:

• \( p(23) = 0.15 \) (from the bar height)
• \( p(24) = 0.05 \) (from the bar height, since the bar for 24 is at 0.05)
• \( p(27) = 0.35 \) (given)
• \( P(24 \text{ or } 26) = p(24) + p(26) = 0.3 \). Since \( p(24)=0.05 \), then \( p(26)=0.3 - 0.05 = 0.25 \)
• Now, let's find \( p(25) \). Wait, the total probability should be 1? Wait, no, the sum of all \( p(x) \) for \( x = 23,24,25,26,27 \) should be 1. So:

\( p(23) + p(24) + p(25) + p(26) + p(27) = 1 \)

Substitute known values:

\( 0.15 + 0.05 + p(25) + 0.25 + 0.35 = 1 \)

Calculate sum of known: \( 0.15 + 0.05 = 0.2; 0.2 + 0.25 = 0.45; 0.45 + 0.35 = 0.8 \). So \( p(25) = 1 - 0.8 = 0.2 \)

Now, \( P(x \leq 25) = p(23) + p(24) + p(25) = 0.15 + 0.05 + 0.2 = 0.4 \)

Wait, let's verify:

• \( p(23)=0.15 \)
• \( p(24)=0.05 \)
• \( p(25)=0.2 \) (calculated)
• \( p(26)=0.25 \) (calculated)
• \( p(27)=0.35 \) (given)

Sum: \( 0.15 + 0.05 + 0.2 + 0.25 + 0.35 = 1 \). Correct.

So \( P(x \leq 25) = p(23) + p(24) + p(25) = 0.15 + 0.05 + 0.2 = 0.4 \)

Step2: Calculate \( P(x \leq 25) \)

\( P(x \leq 25) = p(23) + p(24) + p(25) \)

Substitute the values:

\( p(23) = 0.15 \), \( p(24) = 0.05 \), \( p(25) = 0.2 \)

So \( 0.15 + 0.05 + 0.2 = 0.4 \)

Answer:

\( 0.4 \)