Rasilimali za Kujifunza za Daraja la Kijani Kwa Probability

Muhtasari

Welcome to the course material on Probability in General Mathematics. Probability is a fundamental concept in mathematics that deals with the likelihood of different events occurring. It is widely used in various fields such as statistics, economics, science, and everyday decision-making.

One of the key objectives of this topic is to enable you to solve simple problems in probability, including both addition and multiplication of probabilities. Understanding the basic principles of probability will not only enhance your mathematical skills but also sharpen your analytical thinking and decision-making abilities.

Probability is often represented as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. Events with a probability closer to 1 are more likely to occur, while those closer to 0 are less likely to occur.

When working with probability, it is essential to consider different outcomes and determine their chances of happening. This involves calculating the ratio of favorable outcomes to the total number of outcomes in the sample space.

One of the fundamental concepts in probability is experimental probability, which involves conducting experiments such as tossing a coin, rolling a dice, or picking a card. By observing the outcomes of these experiments, we can calculate the probability of specific events occurring.

Additionally, we will explore the principles of addition and multiplication of probabilities. In probability theory, the addition rule is used to find the probability of the union of two events, while the multiplication rule calculates the probability of the intersection of events.

In this course material, we will delve into topics such as frequency distribution, histograms, bar charts, and pie charts to visually represent data and probabilities. You will also learn about measures of central tendency, including mean, mode, and median, which help summarize data and provide insights into the average and most common values.

Furthermore, we will discuss cumulative frequency, range, mean deviation, variance, and standard deviation to understand the dispersion and variability of data. These statistical measures play a crucial role in analyzing data and making informed decisions based on probabilities.

Overall, mastering the concepts of probability will empower you to make informed predictions, analyze uncertain scenarios, and solve a wide range of problems in various fields. By the end of this course material, you will have a solid foundation in probability theory and the practical skills to apply it in real-world situations.

Malengo

Apply the multiplication rule of probability
Apply the addition rule of probability
Calculate probabilities for simple events
Understand the concept of probability

Maelezo ya Somo

Probability is a branch of mathematics that deals with the likelihood of the occurrence of a given event. It is a measure of the chance that a particular event will happen. Probabilities are expressed as numbers between 0 and 1, where 0 indicates that an event will not occur, and 1 indicates that an event will certainly occur. Understanding probability is essential in various fields such as science, engineering, economics, and everyday life.

Tathmini ya Somo

Hongera kwa kukamilisha somo la Probability. Sasa kwa kuwa umechunguza dhana na mawazo muhimu, ni wakati wa kuweka ujuzi wako kwa mtihani. Sehemu hii inatoa mazoezi mbalimbali maswali yaliyoundwa ili kuimarisha uelewaji wako na kukusaidia kupima ufahamu wako wa nyenzo.

Utakutana na mchanganyiko wa aina mbalimbali za maswali, ikiwemo maswali ya kuchagua jibu sahihi, maswali ya majibu mafupi, na maswali ya insha. Kila swali limebuniwa kwa umakini ili kupima vipengele tofauti vya maarifa yako na ujuzi wa kufikiri kwa makini.

Tumia sehemu hii ya tathmini kama fursa ya kuimarisha uelewa wako wa mada na kubaini maeneo yoyote ambapo unaweza kuhitaji kusoma zaidi. Usikatishwe tamaa na changamoto zozote utakazokutana nazo; badala yake, zitazame kama fursa za kukua na kuboresha.

A box contains 5 red balls, 3 blue balls, and 2 green balls. If a ball is selected at random from the box, what is the probability that it is blue? A. 1/5 B. 1/3 C. 3/10 D. 2/5 Answer: B. 1/3
In a single throw of a fair six-faced dice, what is the probability of getting an odd number? A. 1/6 B. 1/3 C. 1/2 D. 2/3 Answer: C. 1/2
If two coins are tossed simultaneously, what is the probability of getting at least one tail? A. 1/4 B. 1/2 C. 3/4 D. 1 Answer: C. 3/4
A glass jar contains 4 red marbles, 3 blue marbles, and 5 green marbles. What is the probability of drawing a blue marble, replacing it, and then drawing a green marble? A. 9/60 B. 5/12 C. 1/5 D. 1/12 Answer: A. 9/60
Given that a card is drawn at random from a deck of 52 playing cards, what is the probability of getting a red card (hearts or diamonds)? A. 1/13 B. 1/2 C. 1/4 D. 1/26 Answer: B. 1/2
If the probability of an event E occurring is 0.4, what is the probability of the event not occurring? A. 0.6 B. 0.4 C. 2/5 D. 3/5 Answer: A. 0.6
In a class of 25 students, what is the probability that a randomly selected student is either a boy or has black hair if 12 students are boys and 9 students have black hair? A. 13/25 B. 21/25 C. 11/25 D. 5/25 Answer: B. 21/25
If the probability of winning a game is 3/10, what is the probability of losing the game? A. 1/3 B. 1/5 C. 7/10 D. 3/5 Answer: C. 7/10
In a survey, 60% of the people like chocolate ice cream, 30% like vanilla, and 10% like strawberry. What is the probability that a person selected at random does not like chocolate, vanilla, or strawberry ice cream? A. 0.5 B. 0.3 C. 0.4 D. 0.6 Answer: A. 0.5

Vitabu Vinavyopendekezwa

	Elementary Statistics Manukuu Mean, Mode, and Median Simplified Mchapishaji Pearson Education Mwaka 2018 ISBN 978-0134462721
	Introductory Probability and Statistics Explorations Manukuu A Guide to Understanding Data Mchapishaji OpenStax Mwaka 2020 ISBN 978-1719872140

Maswali ya Zamani

Unajiuliza maswali ya zamani kuhusu mada hii yanaonekanaje? Hapa kuna idadi ya maswali kuhusu Probability kutoka miaka iliyopita.

JAMB UTME - Mathematics - 2023 (Bofya kufanya tathmini kamili)

Swali 1 Ripoti

A bag contains 8 red balls and some white balls. If the probability of drawing a white ball is half of the probability of drawing a red ball then find the probability of drawing a red ball and a white ball if the balls are drawn without replacement.

\frac{1}{3}

\frac{2}{9}

\frac{2}{3}

\frac{8}{33}

Maelezo ya Majibu

Let the number of white balls be n

The number of red balls = 8

Now, the probability of drawing a white ball = $\frac{n}{n + 8}$

The probability of drawing a red ball = $\frac{8}{n + 8}$

Since Pr(White ball) = $\frac{1}{2}$ x Pr(Red ball)

$∴ \frac{n}{n + 8} = \frac{1}{2} \times \frac{8}{n + 8}$

= $\frac{n}{n + 8} = \frac{4}{n + 8}$

$∴ n = 4$

Number of white balls = 4

The possible number of outcomes = n + 8 = 4 + 8 = 12

Pr(Red ball and White ball) = Pr(Red ball) x Pr(White ball)

Pr(Red ball) = $\frac{8}{12}$

Pr(Red ball) = $\frac{4}{11}$

Pr(Red ball and white ball) = $\frac{8}{12} \times \frac{4}{11}$

$∴$ Pr(Red ball and white ball) = $\frac{8}{33}$

Tazama Jibu

NECO SSCE - General Mathematics - 2005 (Bofya kufanya tathmini kamili)

Swali 1 Ripoti

Two fair dice are tossed together once. What is the probability of getting a total of at least 9 from the outcome?

$\frac{1}{9}$

$\frac{1}{6}$

$\frac{5}{18}$

$\frac{1}{3}$

$\frac{13}{18}$

Tazama Jibu

WAEC SSCE - General Mathematics - 2006 (Bofya kufanya tathmini kamili)

Swali 1 Ripoti

Two fair dice are tossed together once.

(a) Draw a sample space for the possible outcomes ;

(b) Find the probability of getting a total : (i) of 7 or 8 ; (ii) less than 4.

Jibu

Tazama Jibu