# The probability with equations

Of the three independent events $E_1,E_2,E_3$,the probability that only $E_1$ occurs is $\alpha$,only $E_2$ is $\beta$ and only $E_3$ is $\gamma$. Let the probability $p$ that none of the events occurs satisfy the equations $(\alpha-2\beta)p=\alpha\beta$ and $(\beta-3\gamma)p=2\beta\gamma$. Probability of occurence of $E_1$ is$X$ and probability of occurence of $E_3$ is $Y$. Find $\frac{X}{Y}$. What can i do to proceed?I'm totally strucked!

Let $\delta, \epsilon, \zeta$ be the probabilities that $E_1$ and $E_2$, $E_1$ and $E_3$, and $E_2$ and $E_3$ occur, respectively. Let $\eta$ be the probability that all three occur.
$$p = 1 - \alpha - \beta - \gamma + \delta + \epsilon + \zeta - \eta.$$
Also, $X = \alpha + \delta + \epsilon - \eta,$ and $Y = \beta + \delta + \zeta - \eta.$
{"ft_lang_label":"__label__en","ft_lang_prob":0.7632496,"math_prob":1.0000098,"size":917,"snap":"2023-40-2023-50","text_gpt3_token_len":320,"char_repetition_ratio":0.15005477,"word_repetition_ratio":0.0,"special_character_ratio":0.37186477,"punctuation_ratio":0.14285715,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.00001,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-23T07:44:02Z\",\"WARC-Record-ID\":\"<urn:uuid:25fa6cbd-c1ad-4699-aeef-f4d6151d61a4>\",\"Content-Length\":\"134738\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:be87cfc0-3897-40f5-bcb6-ac0cbe393209>\",\"WARC-Concurrent-To\":\"<urn:uuid:c4ad89f1-690b-487c-abd3-099c054309c9>\",\"WARC-IP-Address\":\"104.18.10.86\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/658666/the-probability-with-equations\",\"WARC-Payload-Digest\":\"sha1:4WRJMXA255Q7CWRN3Y44MRETGXFDVSC7\",\"WARC-Block-Digest\":\"sha1:3QT7LM2UQUGOVEHANHYFPH4JVYXGLQBQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233506480.35_warc_CC-MAIN-20230923062631-20230923092631-00693.warc.gz\"}"}